Blog

The Algebra and Geometry of $\mathbb Q_p$

In Absolute Values and Completions of $\mathbb Q$ we looked at the completions of $\mathbb Q$, and in particular the non-archimedean completions $\mathbb Q_p$, from the viewpoint of analysis and topology. Here we investigate the algebraic and geometric properties of the $p$-adic numbers, though we will not ignore the topology completely.

The archimedean property of the real numbers asserts that for any real number $x$ there is an integer $n$ such that $| x – n |_{\infty} \leq 1/2$. That is, every real number is at most $1/2$ unit away from its closest integer. We contrast this with $\mathbb Q_p$ where $| n |_p \leq 1$ for all $n \in \mathbb Z$, and where we can find $x \in \mathbb Q_p$ with $|x|_p$ as large (or small) as we want. Suffice it to say $\mathbb Q_p$ does not have the archimedean property, hence the adjective non-archimedean for such fields.

The completion of $\mathbb Z$ in $\mathbb Q_p$ is denoted $\mathbb Z_p$ and explicitly given by $$\mathbb Z_p = \{ x \in \mathbb Q_p : |x|_p \leq 1 \}.$$ Put another way, $p$-adic integers are equal to the closed unit ball in $\mathbb Q_p$.

It is easy to verify that $\mathbb Z_p$ is a ring; the only property in doubt is closure under addition, but this comes from the strong triangle inequality. If $x, y \in \mathbb Z_p$ then $$|x+y|_p \leq \max\{ |x|_p, |y|_p\} = 1.$$ This same argument shows that $\mathfrak m_p = \{ x : |x|_{p} < 1 \}$ is itself a ring—in fact a maximal ideal—of $\mathbb Z_p$. We also define $U_p = \{ x : |x|_p = 1\}$. Note that $U_p$ is not a ring, but it is a group under multiplication: the group of units in $\mathbb Z_p$. We note that the definition of $\mathbb Q_p, \mathbb Z_p, \mathfrak m_p$ and $U_p$ are invariant under substitution of an equivalent absolute value. That is, the $p$ that indexes these sets is associated to the place indexed by $p$ not the specific choice of absolute value $| \cdot |_p$.

Commutative rings and maximal ideals quotient to make fields, and we define the residue field of $\mathbb Q_p$ to be $\mathbb F_p \cong \mathbb Z_p/\mathfrak m_p$, which as the notation suggests is the finite field with $p$ elements. We will prove this in the next section.

An explicit construction of $\mathbb Q_p$

We have defined the $p$-adic numbers as equivalence classes of Cauchy sequences. It is useful to have a proscribed choice of representative for each equivalence class. This is done using series. Consider the formal series $$ \sum_{m=v}^{\infty} a_m p^m$$ where each $a_m \in \{0,1,\ldots,p-1\}$. The $M$th partial sum is a rational number $$n_M = \sum_{m=v}^M a_m p^m,$$ and $|n_M – n_{M-1}|_p \leq p^{-M}$. In fact, $$|n_{M + n} – n_M|_p \leq \max_{\ell=1,\ldots n}\{|n_{M+\ell} – n_{M+\ell-1}|_p\} \leq p^{-M}$$ and hence $(n_M)$ is Cauchy with respect to $| \cdot |_p$. It follows that $$ x = \lim n_M = \sum_{m=v}^{\infty} a_m p^m$$ defines a $p$-adic number with $|x|_p \leq p^{-v}$. Note that when $x$ is a positive rational integer, its series representation has finitely many terms (indexed starting at 0) and is simply its base-$p$ expansion.

So every power series of this form produces a $p$-adic number. What about the converse? Given a $p$-adic number, can we find a representative given as a “base $p$ series”?

Theorem

Suppose $x \in \mathbb Q_p$ then there exists an integer $v$ and a sequence of integers $(a_n)_{n=v}^{\infty}$ with each $0 \leq a_n < p$ such that $x$ is represented by the sequence of partial sums of $$ \sum_{n=v}^{\infty} a_n p^n.$$ Moreover each such series defines an element of $\mathbb Q_p$.

Proof

We first do this for sequences of integers. Let $(\ell_m)$ be a sequence of integers Cauchy with respect to $| \cdot |_p$. We may assume, by taking a subsequence if necessary that $|\ell_m – \ell_{m-1}|_p \leq p^{-m}$. We make a new sequence of integers $(n_m)$ by taking the base-$p$ expansion of $\ell_m$ and truncating it at the $m$th term. That is, if $\ell_m = \sum_{j=0}^J a_j p^j$ then $n_m = \sum_{j=0}^m a_j p^j$. We note that $|\ell_m – n_m| \leq p^{-m-1}$, and that \begin{align}|n_m – n_{m-1}|_p &= |n_m – \ell_m + \ell_m – \ell_{m-1} + \ell_{m-1} – n_{m-1}|_p \\ &\leq \max\{|n_m – \ell_m|_p, |\ell_m – \ell_{m-1}|_p, |\ell_{m-1} – n_{m-1}| \} \leq p^{-m}.\end{align} All this is to say that $(\ell_m)$ and $(n_m)$ are equivalent Cauchy sequences. It remains to show that $(n_m)$ is the sequence of partial sums of an infinite base-$p$ expansion. Currently we know that $n_m$ is a polynomial in $p$ of degree $m$ with coefficients in $\{0, 1, \ldots, p-1\}$, but we don’t know if these coefficients agree with those (of degree $<m$) of $n_{m-1}$ for all $m$. That is we need, to verify that for each $m$, $n_m = n_{m-1} + a_{m} p^m$ for some $a_m \in \{0, 1, \ldots, p-1\}$. Because $|n_m – n_{m-1}|_p \leq p^{-m}$ we know that $n_m = n_{m-1} + A_m p^m$ for some integer $A_m$. We may replace $A_m$ with $a_m \in \{0, 1, \ldots, p-1\}$ congruent modulo $p$, and by replacing $n_m$ with $n_{m-1} + a_m p^m$ (if necessary), we find that the series $(n_m)$ is the sequence of partial sums of $$\sum_{m=0}^{\infty} a_m p^m$$ as desired.

We are almost done, we now simply need to show that if we have any rational sequence $(r_m)$ Cauchy with respect to $| \cdot |_p$ that it can be represented by the partial sums of an infinite base-$p$ expansion of the form $$\sum_{m=v}^{\infty} a_m p^m$$ for some $v < 0$. We could bust out the previous analysis, but here we remark that there is some largest (least negative) integer $v$ such that $p^{v} \mathbb Z_p$ contains $(r_n)$. That is $x_m = p^{-v} r_m$ defines a sequence such that $|x_m|_p \leq 1$. Scaling by $p^{-v}$ is continuous, and so $(x_m)$ is a Cauchy sequence. We define $y_m$ to be the degree $m$ truncation of the base-$p$ expansion of $x_n$. This is exactly what we did before by defining $n_m$ in terms of the $\ell_m$ except here we do not know that the $x_m$ are integers—however we just proved that they have (possibly infinite) base-$p$ expansions because they are all in the closure of the integers—this can be truncated to produce the $y_m$. Regardless all the analysis works and we find that $(y_m)$ is equivalent to $(x_m)$ and is the sequence of partial sums of some $$\sum_{m=0}^{\infty} b_m p^m.$$ We then define $$s_m = p^v \sum_{m=0}^{\infty} b_m p^m.$$ Scaling is still continuous, and so $(s_n)$ is Cauchy and equivalent to $(r_n)$ and is the sequence of partial sums of an infinite base-$p$ expansion (allowing for finitely many negative powers of $p$).

So far we have only constructed positive numbers. Negative numbers can be represented in base-$p$ expansion. In particular, $$ -1 = \sum_{n=0}^{\infty} (p-1) p^n.$$ To compute the negative of a generic number $x$ we simply compute $0-x$ base-$p$.

Example

The base-5 expansion of 429 is $429 = 4 \cdot 5^0 + 0 \cdot 5^1 + 2 \cdot 5^2 + 3 \cdot 5^3$. To compute $-429$ we wish to add powers of 5 that always cause us to “carry the one”.

We find $-429 = (1 \cdot 5^0 + 4 \cdot 5^1 + 2 \cdot 5^2 + 1 \cdot 5^3) + 4 \cdot 5^4 + 4 \cdot 5^5 + \cdots$.

We return to our claim about $\mathbb Z_p/ \mathfrak m_p$.

Theorem

$\mathbb Z_p/ \mathfrak m_p$ is isomorphic to the field with $p$ elements.

The proof is now obvious because $\mathfrak m_p = p \mathbb Z_p$ and thus two base-$p$ expansions are the same modulo $\mathfrak m_p$ if and only if they have the same constant coefficient, and hence $\mathbb Z_p/\mathfrak m_p$ is a field with $p$ elements. It is easy to see that, in fact $p^n \mathbb Z_p/p^{n+1} \mathbb Z_p \cong \mathbb F_p$ (for all $n \in \mathbb Z$).

The Geometric Picture of $\mathbb Z_p$

Here we want to think of the coefficients of $x = \sum_{n=0}^{\infty} a_n p^n \in \mathbb Z_p$ not as coefficients of a power series, but as an address. Imagine driving in a strange town of one-way roads, where at each intersection you have $p$ choices of roads ahead of you (numbered in some consistent way using $0,1, \ldots, p-1$). Then by telling you a sequence of numbers $(a_n)$ with $a_n \in \{0, 1, \ldots, p-1\}$ I am giving you instructions to an address at the end of an infinite sequence of roads.

Six Corners in Chicago. Coming into this intersection along the red arrow, there are five choices, labelled $0, 1,2,3,4$ leaving the intersection. This choice of labelling will be convenient for visualizing elements in $\mathbb Z_5$.

This analogy is not very apt, because we allow no loops in our strange city, but the point remains: we may think of the $(a_n)$ as an “address” for $x = \sum a_n p^n \in \mathbb Z_p$. Each $x \in \mathbb Z_p$ has a unique address, and we may visualize the network of roads as a complete, infinite $p$-nary tree.

Schematic of the “roads” in $\mathbb Z_5$ and $\mathbb Z_3$. These are different embeddings of the complete $5$-nary and $3$-nary trees. The points in $\mathbb Z_p$ are the “boundary” of these trees.

What if we drive only part way to an address? Suppose we start down the roads labelled $(3, 4, 2)$ in $\mathbb Z_5$. This finite tuple then gives us the address of a neighborhood—that consisting of all infinite addresses that start $(3, 4, 2, \ldots )$. Note that $3 + 4 \cdot 5 + 2 \cdot 25 = 73$, and so we can think of this neighborhood as the ball of radius $1/125$ around $73$.

The “roads” to $0, 1$ and $-1$ in $\mathbb Z_5$. Slide left to see the road to the neighborhood defined by the finite tuple $(3,4,2)$. Where is $73$ in that neighborhood?

The positive rational integers can be seen inside $\mathbb Z_p$ as the destination of itineraries which eventually have no turns to the left or right. Negative integers follow itineraries that eventually have a clockwise spiral like that of $-1$. In either event we see visually how $\mathbb Z$ (and indeed $\mathbb N$ and $-\mathbb N$ individually) are dense in $\mathbb Z_p$.

A Bijection Between Balls and Cosets

There is a bijection between the balls in $\mathbb Z_p$ of radius $p^{-n}$ and the cosets of $\mathbb Z_p / p^n \mathbb Z_p$. In the schematic for $\mathbb Z_5$ we may think of a ball as one of the naturally appearing pentagons (of any size). This is not quite right, the ball is actually the fractal bits of the boundary of the tree contained in such a pentagon. In general for $\mathbb Z_p$ there would be a similar schematic with the pentagons (and their fractal tree boundaries) replaced with $p$-gons.

The correspondence between balls and cosets when $p = 5$ and $n=1,2$.

This allows us to index balls of radius $p^{-n}$ by the integers $\{0, 1, \ldots, p^n-1\}$. For instance, the neighborhood in $\mathbb Z_5$ indexed by $(3,4,2)$ is exactly the coset $73 + 125 \mathbb Z_5$.

$\mathbb Z_p$ as a Pro-finite Completion

Another way of specifying the directions to a point $x \in \mathbb Z_p$ is to record the neighborhoods one passes through on the way to $x$. By the correlation between neighborhoods and cosets we can identify that point in $\mathbb Z_p$ with a sequence of cosets $(c_n)$ with $c_n \in \mathbb Z_p/p^n \mathbb Z_p$ represented as integers $0 \leq c_n < p^n$. The relationship between $(c_n)$ and the coefficients $(a_m)$ of the base-$p$ expansion of $x$ is $$c_n = \sum_{m=0}^n a_m p^m.$$ Notice the congruence relations $$c_n \equiv c_{n-1} \bmod p^{\ell}, \qquad \ell=1,\ldots,n.$$ More generally, there is an inverse system of projections $\pi_{\ell \leftarrow n}: \mathbb Z/p^n \mathbb Z_p \rightarrow \mathbb Z/p^{\ell} \mathbb Z_p$, for $0 \leq \ell \leq n$, such that for any $\ell \leq m \leq n$, $\pi_{\ell \leftarrow n} = \pi_{\ell \leftarrow m} \circ \pi_{m \leftarrow n}$. We may thus identify $\mathbb Z_p$ with $$\lim_{\leftarrow n} \mathbb Z/p^n \mathbb Z := \left\{ (c_n) \in \prod_{n=1}^{\infty} \mathbb Z/p^n \mathbb Z : c_{\ell} = \pi_{\ell \leftarrow n}(c_n) \mbox{ for all } 1 \leq \ell \leq n \right\}.$$ This set is exactly the pro-finite completion of the inverse system.

 

A geometric representation of the profinite completion view of the $3$-adic integers. Elements in the $\mathbb Z_3$ are not indexed by the “roads” that you take to get there, but by the cosets aka neighborhoods (here represented as disks) you must pass through to get to your destination. The destination is the same, but the data you use to get there is (only slightly) different.

Addition and multiplication in the pro-finite completion are done component wise, and sums and product remain in $\lim_{\leftarrow } \mathbb Z/p^n \mathbb Z$ because the $\pi_{\ell \leftarrow n}$ are ring homomorphisms. These operations are the same addition and multiplication that comes from the base-$p$ expansion representation of $\mathbb Z_p$.

0

Absolute Values and Completions of $\mathbb Q$

An absolute value on a field $K$ is a function $|\cdot| : K \rightarrow [0, \infty)$ such that for any $x, y \in K$,

  • $|x| = 0$ if and only if $x = 0$;
  • $|x y| = |x| |y|$;
  • $|x + y| \leq |x| + |y|$

These properties are called respectively positive definiteness, multiplicativity and the triangle inequality. If the absolute value satisfies the stronger condition (called the strong triangle inequality)

  • $| x + y| \leq \max\{ |x|, |y| \}$

we say it is a non-archimedean absolute value. An absolute value that does not satisfy the strong triangle inequality is called an archimedean absolute value.

The usual absolute values on $\mathbb Q$, $\mathbb R$ and $\mathbb C$ are all archimedean absolute values.

Every field has a trivial absolute value given by $| 0 | = 0$ and $| x | = 1$ for all $x \neq 0$. This absolute value is not very interesting and we will usually concentrate on the non-trivial absolute values of a field.

Absolute Values on $\mathbb Q$

To distinguish the usual absolute value from new ones we may construct we will denote it $| \cdot |_{\infty}$. That is $$ |x|_{\infty} = \left\{ \begin{array}{rl} x & x \geq 0; \\ -x & x < 0. \end{array}\right.$$

If $p$ is a prime integer then define the valuation $v_p : \mathbb Q^{\times} \rightarrow \mathbb Z$ by $$ v_p(x) = v \qquad x = p^v \frac{a}{b}, \quad \mathrm{GCD}(a, b) = 1.$$ That is, we determine the valuation of a rational number by determining the highest power of $p$ that divides it. This power is positive if the numerator has $p$ as a factor, and is negative if the denominator has $p$ as a factor (when written in lowest terms). If the valuation is 0 then the rational number does not have $p$ in its factorization.

The valuation $v_p$ is a homomorphism from $(\mathbb Q^{\times}, \cdot) \rightarrow (\mathbb Z, +)$. That is $v_p$ is additive: $v_p(xy) = v_p(x) + v_p(y)$. It is common to take $v_p(0) = \infty$ with the justification that $0$ is “infinitely divisible” by $p$.

We may write a non-zero rational number $x$ in terms of the various valuations $\{v_p(x) : p \mbox{ prime} \}$ by $$x = \prod_p p^{v_p(x)}.$$ Note that $v_p(x) \neq 0$ for only the primes that appear in the factorization of $x$. It follows that this product is actually a finite product.

We define the $p$-adic absolute value on $\mathbb Q$ by $$| x |_p = p^{-v_p(x)}.$$ Of course, we need to verify that this is an absolute value. Positive definiteness is a matter of definition, multiplicativity comes from the additivity of $v_p$. Only the triangle inequality remains, and we will in fact show that $| \cdot |_p$ satisfies the strong triangle inequality. Suppose $x = p^v a/b$ and $y = p^u c/d$ where $a/b$ and $c/d$ are written in lowest terms. Then $$ x + y = \frac{p^v d a + p^u b c}{bd}.$$ Because $bd$ is relatively prime to $p$, we see that the minimum power of $p$ we can pull out of the numerator is $\min\{u, v\}$. That is $v_p(x + y) \geq \min\{v_p(x), v_p(y) \}$. This is equivalent to the strong triangle inequality.

Note that we actually proved something slightly stronger than the strong triangle inequality here. Written in terms of valuations and absolute values, $$v_p(x + y) > \min\{v_p(x), v_p(y) \} \quad \mbox{only if} \quad v_p(x) = v_p(y)$$ $$|x + y|_p < \max\{|x|_p, |y|_p \} \quad \mbox{only if} \quad |x|_p = |y|_p.$$

Thus, the $p$-adic absolute value is a non-archimedean absolute value.

The Places of $\mathbb Q$

We say two absolute values $| \cdot |_0$ and $| \cdot |_1$ on a field $K$ are equivalent if there is a positive real number $c$ such that $|\cdot |_0 = |\cdot|_1^c$. It is easily verified that this gives an equivalence relation on the set of absolute values of $K$, and we call the equivalence classes the places of $K$. The place corresponding to the trivial absolute value (which is the only representative in its class) is called the trivial place, and is often excluded from attention.

We will eventually talk about how to complete $K$ with respect to an absolute value, using the same methods as when we construct $\mathbb R$ out $\mathbb Q$ with respect to the usual absolute value. We will see that equivalent absolute values produce the same completion, and different places produce different completions. The completion of $\mathbb Q$ with respect to the trivial absolute value is $\mathbb Q$ itself—another hint that nothing interesting happens with trivial absolute values.

The set of non-trivial places of $K$ is denoted $\mathcal M_K$.

Ostrowski’s Theorem

If $| \cdot |$ is a non-trivial absolute value on $\mathbb Q$ then $|\cdot|$ is equivalent to either the usual absolute value $| \cdot |_{\infty}$ or is equivalent to $| \cdot |_p$ for some prime $p$. That is $\mathcal M_{\mathbb Q}$ is in correspondence with $\mathcal P = \{ \mbox{ primes } \} \cup \{ \infty \}$.

For a proof, see https://en.wikipedia.org/wiki/Ostrowski%27s_theorem#Proof

It is hard to understate the importance of this result. It is surprising (though less so after reading the proof) to see a correspondence between the primes and a set of analytic object. It suggests (rightly) that there is progress to be made in understanding the primes by understanding absolute values. Indeed, as we will see this correspondence extends to number fields, and we will see a bijection between non-archimedean absolute values and prime ideals. The archimedean absolute values in that setting correspond to the real and complex embeddings of the number field. It all hangs together very nicely.

The Product Formula

The product formula is the trivial observation that for any rational number $x \neq 0$, $$ \prod_{p \in \mathcal P} | x |_p = 1.$$

This can be verified by calculation: $$ | x |_{\infty} = \prod_{p \mbox{ prime} } p^{v_p(x)} \quad \mbox{and} \quad \prod_{p \mbox{ prime}} | x |_p = \prod_{p \mbox{ prime}} p^{-v_p(x)},$$ and the product formula follows.

In spite of its triviality, it is an important observation and we will return to a version of the product formula for number fields shortly.

Completions

Recall these two definitions from elementary analysis.

Definition

A sequence of rational numbers $(x_n)$ converges to 0 with respect to the absolute value $| \cdot |$ if, for all $\epsilon > 0$ there exists positive integer $N$ such that if $n \geq N$ then $|x_n| < \epsilon$. In this situation we write $\lim x_n = 0$ or $(x_n) \rightarrow 0$.

Definition

A sequence of rational numbers is Cauchy with respect to $| \cdot |$ if, for all $\epsilon > 0$ there exists positive integer $N$ such that if $n, m \geq N$ then $|x_n – x_m| < \epsilon$. We will denote the set of Cauchy sequences by $\mathcal C$.

If you recall from elementary analysis, Cauchy sequences are convergent sequences. However, not every convergent sequence of rational numbers converges to a rational number. That is, taking limits can take you out of $\mathbb Q$. These new limit points live in the completion, and that completion depends on the absolute value (in fact place) used in the definition of convergence and Cauchy.

The equivalence of Cauchy sequences and convergent sequences (once you account for new limit points) is useful because the condition of Cauchyness depends only on the rational numbers in the sequence. That is we can determine if something is Cauchy without having to know what its limit is or where that limit lives.

The other thing that is useful about Cauchy sequences, is that they form a ring under coordinate-wise addition and multiplication. This is essentially equivalent to the limit laws: If $(x_n), (y_n) \in \mathcal C$ then $(x_n + y_n) \in \mathcal C$ and $(x_n y_n) \in \mathcal C$. The identically zero sequence $(0)$ is the additive identity, and $(1)$ is the multiplicative identity. Cauchyness of the coordinate-wise quotient of two sequences also follows from the limit laws, though one has to be careful that the Cauchy sequence in the denominator does not converge to 0, and to “throw out” any quotients in the sequence where the denominator may be 0. We may embed $\mathbb Q \hookrightarrow \mathcal C$ by sending $x$ to the constant sequence $(x)$.

In the real numbers there are many different sequences of rational numbers which converge to the same real number. For instance, much mathematics has been made of discovering interesting rational sequences that converge to some of our favorite irrational numbers, like $\pi$. Notice if we do have two sequence $(x_n)$ and $(y_n)$ and $(x_n) \rightarrow \pi$ and $(y_n) \rightarrow \pi$, and here $\pi$ can be replaced by any real number, then $(x_n – y_n) \rightarrow 0$. Thus we can determine when two convergent sequences converge to the same number, simply by determining whether their difference converges to 0.

Returning to $\mathcal C$, we define an equivalence relation $(x_n) \equiv (y_n)$ if $(x_n – y_n) \rightarrow 0$. In our head we should think of an equivalence class as the set of all Cauchy sequences that converge to the same number, and there is one equivalence class for every possible limit point. And indeed, that is the definition of the completion of $\mathbb Q$ with respect to $| \cdot |$. It is the field formed from the equivalence classes of Cauchy sequences. We add, subtract, multiply and divide by choosing representatives of the equivalence classes and performing the appropriate operation coordinate-wise, and returning the equivalence class of that new Cauchy sequence. The rational number $x$ is represented by the equivalence class of the constant sequence $(x)$.

Let us denote the completion by $\overline{\mathbb Q}$ (this obviously depends on the absolute value, but for the moment we are working the distinguished absolute value $| \cdot |$). Suppose we have another equivalent absolute value $| \cdot |_0 = | \cdot |^c; c > 0$. We wish to argue that both absolute values produce the same completion, that is that the completion is something more appropriately associated to a place rather than a single absolute value. This is done by showing that the absolute values $| \cdot |$ and $| \cdot |_0$ determine the same set of Cauchy sequences, and that they determine the same equivalence relations on those Cauchy sequences.

Suppose $(x_n)$ is Cauchy, and given $\epsilon$ let $N(\epsilon)$ be the guaranteed integer such that $n, m \geq N(\epsilon)$ implies $| x_n – x_m | < \epsilon$. It follows that if we set $N = N(\epsilon^{1/c})$ then $| x_n – x_m |_0 = |x_n – x_m|^c < |\epsilon^{1/c}|^c = \epsilon$. Thus if $(x_n)$ is Cauchy with respect to $| \cdot |$ it is Cauchy with respect to $| \cdot |_0$ the reverse containment is established similarly (but setting $N = N(\epsilon^c)$), and we see both absolute values produce the same Cauchy sequences.

We also need to establish that $(x_n) \rightarrow 0$ with respect to $| \cdot |$ if and only if it does the same with respect to $| \cdot |_0$. The argument is almost identical to that used to show that both absolute values produce the same Cauchy sequences.

The places are in correspondence with $\mathcal P$ and we denote the completion with respect to $| \cdot |_p$ by $\mathbb Q_p$. In particular $\mathbb Q_{\infty}$ is the real numbers.

Extending Absolute Values to $\mathbb Q_p$

Given an element in $\mathbb Q_p$ as represented by Cauchy sequence $x = (x_n)$, then we define $|x|_p = \lim |x_n|_p$ where we take the limit in the real numbers as usual. Upon showing this is well-define, we arrive at an absolute value (which we continue to denote $| \cdot |_p$) on $\mathbb Q_p$ which restricts to $| \cdot |_p$ on $\mathbb Q$.

Suppose $x = (x_n)$ and $y = (y_n)$ are in the same equivalence class—that is $\lim |x_n – y_n|_p = 0$. Then $$| x |_p = \lim | x_n |_p = \lim |y_n + (x_n -y_n)| _p \leq \lim |y_n|_p + \lim |x_n – y_n|_p = |y|_p.$$ But symmetry implies that $|y|_p \leq |x|_p$ so in fact $|x|_p = |y|_p$ and $| \cdot |$ is well defined on $\mathbb Q_p.$

To verify $| \cdot |_p$ is an absolute value on $\mathbb Q_p$, notice that if $| x |_p = 0$ then $(x_n)$ is equivalent to the series $(0)$, that is $x$ is in the zero equivalence class. Multiplicativity follows from the multiplication limit law for series. The triangle inequality likewise follows since $$ | x + y |_p = \lim |x_n + y_n|_p \leq \lim |x_n|_p + \lim|y_n|_p = |x|_p + |y|_p$$ as does the strong inequality (via the continuity of the max function) in the case $| \cdot |_p$ is non-archimedean.

Equipping a field with an absolute value also allows us to define distances, and hence a metric topology. This topology in turn generates a $\sigma$-algebra (which is equal to the Borel $\sigma$-algebra when $p = \infty$) and translation invariant measures (a la Lebesgue measure) on $(\mathbb Q_p, +)$ and $(\mathbb Q^{\times}_p, \cdot)$. I am getting ahead of myself, but the point is that $\mathbb Q_p$ isn’t just a new field with few limit points filled in from $\mathbb Q$, but rather it is a metric space and a measure space and exhibits many features in common with $\mathbb R$.

Different Places Produce Different Completions

Before digging into the topology of $\mathbb Q_p$ we want to justify out claim that the completions of $\mathbb Q$ are in correspondence with $\mathcal P$. So far we have seen that every completion is equal to $\mathbb Q_p$ for some $p \in \mathcal P$, but we have not yet established that different elements in $\mathcal P$ produce different completions.

This is a consequence of the Weak Approximation Theorem for $\mathbb Q_p$, which in our case says that if $| \cdot |$ and $| \cdot |_0$ are non-equivalent absolute values, then you can find an element $x \in \mathbb Q$ such that $| x |$ is large and $| x |_0$ is small (and vice-versa). This will in term imply that different sequences are Cauchy or converge to 0 with respect to these different absolute values.

The Weak Approximation Theorem

Suppose $p_1, \ldots, p_N \in \mathcal P$ index any finite number of places of $\mathbb Q$ and $1 \leq n < N$. Given $a \in \mathbb Q$ and any $\epsilon > 0$ there exists $x \in \mathbb Q$ such that $|x – a|_{p_1}, \ldots, |x – a|_{p_n} \in (0, \epsilon)$ and $|x-a|_{p_{n+1}}, \ldots, |x-a|_{p_N} \in [\epsilon^{-1}, \infty).$

We do this for $a=0$ as follows, first we suppose the $p_1, \ldots, p_N$ are all non-archimedean absolute values and then discuss how to modify the argument if we also want $| x |_{\infty}$ either large or small. Consider $$ x = \frac{p_1^{m_1} \cdots p_n^{m_n}}{p_{n+1}^{m_{n+1}} \cdots p_N^{m_N}}. $$ Then $|x|_{p_1} = p_1^{-m_1}, \ldots, |x|_{p_n} = p_n^{-m_n}$ and these can be made as small as desired by choosing $m_1, \ldots, m_n$ as large as necessary. Similarly, $|x|_{p_{n+1}} = p_{n+1}^{m_{n+1}}, \ldots |x|_{p_N} = p_N^{m_N}$ which can be made as large as desired by choosing $m_{n+1}, \ldots, m_N$ as large as necessary.

How would we simultaneously make $|x|_{\infty}$ as large or small as specified? We may multiply $x$ by a rational number $r$ whose factorization avoids $p_1, p_2, \ldots, p_N$ without changing any of the absolute values $|\cdot|_{p_1}, \ldots, |\cdot|_{p_N}$. That is $|r x|_{p_1} = |x|_{p_1}, \ldots, |r x|_{p_N} = |x|_{p_N}$. Note however, that $|r x|_{\infty} = |x|_{\infty} |r|_{\infty}$, and hence by choosing $r$ sufficiently large or small, the rational number $r x$ will satisfy the conclusions of the theorem when $a=0$. The general case is a consequence of the Chinese Remainder Theorem.

Completions as Topological and Measurable Spaces

Here we are mostly going to assume that $p < \infty$, though we will compare and contrast the situation with the $\mathbb Q_{\infty} = \mathbb R$ case.

The Borel Topology on $\mathbb Q_p$

Once we have an absolute value on a field we can make $\epsilon$-neighborhoods, and these form the basis for a topology. In the case of $x \in \mathbb Q_p$, given $\epsilon > 0$, we define $$B_{\epsilon}(x) = \{ y \in \mathbb Q_p : |y – x|_p < \epsilon \}.$$ When $p = \infty$ these are the usual neighborhoods of $x \in \mathbb R$. The topology generated by all such sets is called the Borel topology. It is easy to see that the collection of open neighborhoods does not depend on which absolute value you use from a particular place. An $\epsilon$-neighborhood with respect to $| \cdot |^c$ is an $\epsilon^{1/c}$-neighborhood of $| \cdot |$ and thus the set of neighborhoods is the same. So from the topological point of view $\mathbb Q_p$ is more naturally associated to a place than to a specific absolute value.

When $p < \infty$ something interesting happens that does not happen in $\mathbb R$. First note that, unlike $| \cdot |_{\infty}$, the non-archimedean absolute values are discrete. Namely $| \cdot |_p$ takes values in $\{p^n : n \in \mathbb Z\}$. This means that any open ball $B_{\epsilon}(x)$ can also be described as a closed ball $\overline{B}_{\epsilon’}(x)$ for some slightly larger $\epsilon’ > \epsilon$. The language sometimes used is that the balls in $\mathbb Q_p$ are clopen.

The balls in $\mathbb Q_p$ are nested in a way that they are not for $\mathbb R$. Namely two balls in $\mathbb Q_p$ are either disjoint, or one is a subset of the other. That is, for $p < \infty$, $\mathbb Q_p$ is totally disconnected. This, like all other differences is driven by the strong triangle inequality. To see this, suppose $B_{\epsilon}(x)$ and $B_{\delta}(y)$ are balls in $\mathbb Q_p$ with $z \in B_{\epsilon}(x) \cap B_{\delta}(y)$. Without loss of generality we may assume $\epsilon \leq \delta$.

First note $x \in B_{\delta}(y)$: $|x – z|_p < \epsilon$ and $|z – y|_p < \delta$. It follows that $|x-y|_p \leq \max\{|x-z|_p, |z-y|_p\} = \delta$.

If $x \not \in B_{\delta}(y)$ then the strong triangle inequality is violated (dotted distance in purple).

Next, if $w \in B_{\epsilon}(x)$ then $|w-y|_p \leq \max\{|w – x|_p, |x – y|_p\} = \delta$ and hence $w \in B_{\delta}(y)$ as claimed. It follows that $B_{\epsilon}(x) \subset B_{\delta}(y)$.

If $w \not \in B_{\delta}(y)$ then the strong triangle inequality is violated (dotted distance in red).

Another property of $\mathbb Q_p$ (this time shared between $p$ finite and infinite) is local compactness. Recall the definition: a space is locally compact if every point $x$ has a neighborhood which is contained in a compact set. It turns out that $\mathbb Q_p$ has the Heine-Borel property, and for any $x$, there is an epsilon such that $x \in B_{\epsilon}(x) \subset \overline B_{\epsilon}$ which does the job.

Haar measure on $\mathbb Q_p$

In the standard manner we set $\mathcal B$ to be the $\sigma$-algebra on $\mathbb Q_p$ generated by all balls. When $p < \infty$ the total disconnectivity of $\mathbb Q_p$ means that the generic sets in $\mathcal B$ are much easier to describe that in the $\mathbb R = \mathbb Q_{\infty}$ situation. Namely, because the countable intersection of balls is either another ball or a singleton (a set containing a single point), we see that a generic set in $\mathcal B$ looks like a countable union of balls and singletons. This is very tidy in comparison the nightmarishness of a general Borel subset of $\mathbb R$.

$(\mathbb Q_p, +)$ and $(\mathbb Q_p^{\times}, \cdot)$ are locally compact abelian groups. Locally compact abelian groups are important kinds of topological and measurable spaces because they can be equipped with a translation invariant measure. Specifically, if $B \in \mathcal B$ is a Borel set and $x \in \mathbb Q_p$ then we call $$ x + B = \{ x + b : b \in B \} \qquad \mbox{and} \qquad x B = \{ x b : b \in B\}$$ the additive and multiplicative translations of $B$ by $x$. A measure $\mu$ on $(\mathbb Q_p, \mathcal B)$ is said to be translation invariant if $\mu(x + B) = \mu(B)$ for all $B \in \mathcal B$ and all $x \in \mathbb Q_p$. In the case of $(\mathbb Q_p^{\times}, \mathcal B^{\times})$ the corresponding condition for the multiplicactive translation invariant measure $\mu^{\times}$ is $\mu^{\times}(x B) = \mu^{\times}(B)$. (I have not formally introduced the $\sigma$-algebra, $\mathcal B^{\times}$ on $\mathbb Q_p^{\times}$ but as as a set of points $\mathbb Q_p^{\times}$ is simply $\mathbb Q_p$ with 0 removed. We may take $\mathcal B’$ to be the $\sigma$-algebra generated by all balls except those containing 0.)

We may make $\mu$ and $\mu^{\times}$ unique by specifying the measure of a single clopen set (or in the case of $\mathbb Q_{\infty} = \mathbb R$ a single (non-singleton) closed interval). Thus we define the measures $\mu_p$ and $\mu^{\times}_p$ to be the unique translation invariant measures on $(\mathbb Q_p, \mathcal B)$ and $(\mathcal Q_p^{\times}, \mathcal B^{\times})$ normalized so that $$ \mu_p \{ x : |x|_p \leq 1 \} = 1 \qquad \mbox{and} \qquad \mu_p^{\times} \{x : |x|_p = 1 \} = \frac{p-1}{p}.$$ These measures are referred to as the (normalized) Haar measures for $\mathbb Q_p$ and $\mathbb Q_p^{\times}$.

The normalization on $\mu_p^{\times}$ may look a little strange. This choice was motivated by the fact that for any $x \in \mathbb Q_p$ and $B \in \mathcal B$, $$\mu_p(x B) = |x|_p \mu_p(B).$$ This implies that $$\mu_p^{\times}(dx) = \frac{\mu_p(dx)}{|x|_p}.$$ Moreover, if we set $C$ to be the closed unit ball, we have $p C = \{ x \in \mathbb Q_p : |x|_p < 1 \}$ and $\mu(pC) = \mu(C)/p$. It follows that $$\mu_p\{ x : |x|_p = 1 \} = \mu_p(C) – \mu_p(pC) = \frac{p-1}{p}.$$ The normalization for $\mu_p^{\times}$ makes it equal to $\mu_p$ on $\{ x : |x|_p = 1 \}$, a situation which can be advantageous when both measures come into play.

Example

Let $B = \{ x : 0< |x|_p < 1\}$ and $\overline B = \{ x : 0< |x|_p \leq 1\}$, and let $U = \overline B \setminus B = \{x : |x|_p = 1\}$. Then $$ \overline B = U \sqcup B \quad \mbox{and} \quad B = p\overline{B}.$$ Induction then implies that $$\overline B = \bigsqcup_{n=0}^{\infty} p^n U; \qquad \mbox{Indeed} \qquad \mathbb Q^{\times}_p = \bigsqcup_{n \in \mathbb Z} p^n U.$$ This is the decomposition of $\mathbb Q^{\times}_p$ into sets of equal absolute value. That is $p^n U$ is exactly the set where on which $|x|_p = p^{-n}$.

Suppose $s > 0$ and consider $$ \int_{\overline B} |x|_p^s \, \mu_p^{\times}(dx) .$$ Using the decomposition, $$\int_{\overline B} |x|_p^s \, \mu_p^{\times}(dx) = \sum_{n=0}^{\infty} \int_{p^n U} |x|_p^s \, \mu_p^{\times}(dx).$$ The integrand is constant (and equal to $p^{-n s}$) on $p^n U$, and $\mu_p^{\times}(p^N U) = \mu_p^{\times}(U) = (p-1)/p$. Hence, $$\int_{\overline B} |x|_p^s \, \mu_p^{\times}(dx) = \sum_{n=0}^{\infty}p^{-ns} \left(\frac{p-1}{p}\right) = \left(\frac{p-1}{p}\right) \frac{1}{1 – p^{-s}}.$$ This is an important calculation in the theory of the Riemann $\zeta$-function.

0

Field Extensions and Number Fields

Here I am storing various basic facts about Number Fields that are useful in other notes. I hope this becomes more complete as time goes on.

Number Fields

Recall that a number field $K$ is a finite extension of $\mathbb Q$. While we often think of number fields as $\mathbb Q(\alpha)$ for some algebraic number embedded in $\mathbb C$ it is useful to recall the general (unembedded) construction. $\mathbb Q[x]$ is the ring of polynomials with rational coefficients in the indeterminant $x$. If $f(x) \in \mathbb Q[x]$ is irreducible, then $f(x) \mathbb Q[x]$, the ideal formed from all rational polynomials divisible by $f(x)$, is a maximal ideal in $\mathbb Q[x]$. It follows that $K = \mathbb Q[x]/f(x) \mathbb Q[x]$ is a commutative ring with all non-zero elements invertible—that is a field.

In this construction, the elements of $K$ are cosets of the form $g(x) + f(x) \mathbb Q[x]$. If $g(x)$ and $h(x)$ generate the same coset, then we will write $g(x) \equiv h(x)$ (or $g(x) \equiv h(x) \bmod f(x)$ if more clarity is necessary). In this situation $f(x) | (g(x) – h(x))$.

Given the coefficients of $f(x)$, the arithmetic in $K$ is easy to perform. Suppose for $a_0, \ldots, a_{d-1}$ are the rational coefficients to $$f(x) = x^d + \sum_{n=0}^{d-1} a_n x^n,$$ then, $$ x^d \equiv -a_0 – a_1 x – \cdots – a_{d-1} x^{d-1}.$$ Now suppose $g(x) + f(x) \mathbb Q[x]$ is an arbitrary coset. By replacing monomials $x^n$ in $g(x)$ when $n > d$ (serially, if necessary) using this congruence, we see that $g(x) \equiv h(x)$ for some $h(x) \in \mathbb Q[x]$ with $\deg(g) < d$. The polynomial $h(x)$ is equivalent to the result of the Division Algorithm in $Q[x]$ for the remainder of $g(x)$ when divided by $f(x)$.

That is, as a group (in fact, as a vector space) $K$ is isomorphic to $\mathbb Q^d$ where the isomorphism is given by $$(b_0, \ldots, b_{d-1}) \mapsto x^d + \sum_{m=0}^{d-1} b_m x^m + f(x) \mathbb Q[x].$$ The only thing missing in this description is the multiplication. If we want to multiply two vectors $\mathbf b, \mathbf c \in \mathbb Q^d$, we set $g(x)$ to be the monic polynomial with coefficient vector $\mathbf b$ and $h(x)$ to be the polynomial with coefficient vector $\mathbf c$. We first multiply $g(x)$ and $h(x)$ as usual in $\mathbb Q[x]$, and then we use the equivalence $ x^d \equiv -a_0 – a_1 x – \cdots – a_{d-1} x^{d-1}$ to replace monomials in $g(x) h(x)$ (repeatedly if necessary) until we arrive at a polynomial $p(x)$ of degree $< d$. The coefficient vector of this polynomial in $\mathbb Q^d$ is the product of $\mathbf b$ and $\mathbf c$.

$K$, What is it Good for?

First, note that $\mathbb Q \hookrightarrow K$ by the map $r \mapsto r + f(x) \mathbb Q[x]$, and by definition (the fact that $K$ is a vector space of dimension $d$ over $\mathbb Q$) it is a number field of degree $d$ over $\mathbb Q$. This implies $\mathbb Q[x] \hookrightarrow K[x]$, and in particular, $f(x)$ has a life in $K[x]$. Because $f(x)$ is irreducible in $\mathbb Q[x]$ it has no zeroes in $\mathbb Q$. However, we will show that this is no longer the case in $K[x]$. And that is what $K$ is good for—producing a number field where $f(x)$ has a zero.

The element $x + f(x) \mathbb Q[x]$ is the root of $f(x)$ in $K$. To see this, we need only calculate $$f(x + f(x) \mathbb Q[x]) = f(x) + f(x) \mathbb Q[x] = 0 + f(x) \mathbb Q[x].$$

The element $x + f(x) \mathbb Q[x]$ is important as well because if we know how to multiply by this element, then we know how to multiply by arbitrary elements (which are, after all, simply linear combinations of its powers).

Multiplication by $x$ is a linear operator on $\mathbb Q[x]$, indeed $x( a g(x) + h(x) ) = a x g(x) + x h(x)$, and multiplication by $x + f(x) \mathbb Q[x]$ is a linear operator on $K$. We know $K$ is a vector space with basis $( x^n + f(x) \mathbb Q[x] : n=0,\ldots, d-1)$, so it makes sense to talk of the matrix of the multiplication operator, call it $T$, with respect to this basis. Note that, if we denote the standard basis of $\mathbb Q^d$ (with coordinates indexed from 0 to $d-1$ for consistency) by $\mathbf e_0, \ldots, \mathbf e_{d-1}$, then for $n < d-1$, $T \mathbf e_{n} = \mathbf e_{n+1}$. This corresponds to the multiplication $x x^{n} = x^{n+1}$ which remains true in $K$ if $n < d-1$. The final calculation, using the same equivalence that has gotten us so far $ x^d \equiv -a_0 – a_1 x – \cdots – a_{d-1} x^{d-1}$, shows that $T \mathbf e_{d-1} = -a_0 \mathbf e_0 – a_1 \mathbf e_1 – \cdots – a_{d-1} \mathbf e_{d-1}$. It follows that the matrix of $T$ with respect to the basis $(\mathbf e_n)$ is $$ \begin{pmatrix} 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & \cdots & 0 & – a_1 \\ 0 & 1 & \cdots & 0 & – a_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & -a_{d-1}\end{pmatrix}.$$

If this matrix looks familiar it is because it is the (Frobenius) companion matrix to $f(x)$ and the characteristic polynomial of this matrix (and hence the operator $T$) is $f(x)$. Indeed, the irreducibility of $f(x)$ implies that the minimal polynomial of $T$ is $f(x)$ as well.

But $f(x)$ has roots in $\mathbb C$. What about them?

The Fundamental Theorem of Algebra (ironically a theorem in analysis) guarantees that $f(x)$ has $d$ roots (counting multiplicity) in $\mathbb C$. How are they related to the root of $f(x)$ is $K$?

Let’s start with our favorite $\alpha \in \mathbb C$ such that $f(\alpha) = 0$. We know that $\alpha$ is either in $\mathbb R$ or it has a complex conjugate—more about that later. We can embed $K$ into $\mathbb C$ by sending $x + f(x) \mathbb Q[x] \rightarrow \alpha$. That is, $$ a_{d-1} x^{d-1} + \cdots + a_1 x + a_0 + f(x) \mathbb Q[x] \quad \mapsto \quad a^{d-1} \alpha^{d-1} + \cdots a_1 \alpha + a_0.$$ We denote this embedding by $\mathbb Q(\alpha) \subset \mathbb C$. Notice that if $\alpha \in \mathbb R$, then $\mathbb Q(\alpha) \subset \mathbb R$ and we call it a real embedding of $K$.

A count of the real and complex embeddings products the first of the classical invariants of a number field.

Invariants: Number of real and complex embeddings $r_1$ and $r_2$

Let us distinguish the real and complex roots of $f(x)$ by setting $\alpha_1, \ldots, \alpha_{r_1}$ to be the real roots and $\beta_1, \overline{\beta_1}, \ldots, \beta_{r_2}, \overline{\beta_{r_2}}$ be the non-real complex roots. Clearly $r_1 + r_2 = d$. Then the embeddings $\mathbb Q(\alpha_1), \ldots, \mathbb Q(\alpha_{r_1}) , \mathbb Q(\beta_1) , \ldots, \mathbb Q(\beta_{r_2})$ are called the archimedean embeddings of $K$.

The Norm and Trace

Here we wish to work in some generality and consider field extension $K | k$ where both are number fields. Little generality is lost by keeping the example $k = \mathbb Q$ at the front of your mind. However, as many properties of number fields ‘factor through’ intermediate fields (for instance $[ K : \mathbb Q] = [K : k] [k : \mathbb Q]$) it is useful to maintain some generality in notation etc.

We will also abandon our attempt to denote elements of $K | k$ as cosets in $k[x] / f(x) k[x]$, writing for instance $\alpha, \beta, \gamma, \ldots $ for generic field elements. Often we will implicitly identify $K$ with $k(\alpha)$ for some algebraic number $\alpha$ of degree $d$ over $k$. In this situation $\{1, \alpha, \ldots, \alpha^{d-1} \}$ is a basis for $K$, and the matrix of multiplication by $\alpha$ with respect to this basis is exactly the matrix of $T$ as before (the Frobenius companion matrix of the minimal polynomial of $K | k$).

More generally, given any $\gamma \in K$, we can make the linear operator “multiplication by $\gamma$” $T_{\gamma}$. If $\gamma$ is given as a $k$-linear combination of $\{1, \alpha, \ldots, \alpha^{d-1}\}$ then it is relatively easy to compute the matrix of $T_{\gamma}$ with respect to this basis. Note that this matrix has entries in $k$.

Norm and trace of $K|k$

The norm $N_{K|k} : K \rightarrow k$ and trace $\mathrm{Tr}_{K|k} : K \rightarrow k$ are respectively the determinant and trace of $T_{\gamma}$.

This definition is independent of basis, but can be computed explicitly in the basis $\{1, \alpha, \alpha^{d-1}\}$.

If $\beta, \gamma \in K$, then $T_{\beta \gamma} =T_{\beta} \circ T_{\gamma}$ and $T_{\beta + \gamma} = T_{\beta} + T_{\gamma}$. The multiplicativity of the determinant and the additivity of the trace imply that $$N_{K|k}(\beta \gamma) = N_{K|k}(\beta) N_{K|k}(\gamma)$$ and $$\mathrm{Tr}_{K|k}(\beta + \gamma) = \mathrm{Tr}_{K|k}(\beta) + \mathrm{Tr}_{K|k}(\gamma).$$

The norm is a natural homomorphism from $K^{\times}$ onto $k^{\times}$ and the trace is a natural homomorphism from the additive group $(K, +)$ onto $(k,+)$.

0

Recalling Galois Theory

This is a brief reminder of the main ideas of Galois theory. Any proofs purported here are meant to be suggestive. I learned Galois theory out of Dummit and Foote, which I thought was pretty good. I also have Classical Galois Theory by Gaal on my shelf. This book is in essence one giant worksheet. I have not completed many of the exercises, but I suspect anyone who did would gain a remarkable intuition as to how the theory hangs together.

At any rate, this is mostly for me, since I seem to need to be reminded of the basics of Galois theory every few years.

It may be useful to review Field Extensions and Number Fields before continuing.

Automorphisms of Field Extensions

We want to work in a bit of generality here, so we assume $K | k$ is an extension of number fields. Little generality is lost at this point if you take $k = \mathbb Q$.

Definition

An isomorphism of K is called an automorphism. The set of automorphisms of $K,$ $\mathrm{Aut}(K)$, forms a group under composition. An automorphism σ is said to fix $k$ if $\sigma \gamma=\gamma$ for all $\gamma \in k$. The set of automorphisms of $K$ which fix $k$ is denoted $\mathrm{Aut}(K|k)$ and is a subgroup of $\mathrm{Aut}(K)$.

$\mathrm{Aut}(K|k)$ is a finite group of degree at most $d = [K:k]$. We will use this fact, though the proof would take us too far afield.

Automorphisms of $K$ which preserve $k$ permute roots of polynomials with coefficients in $k$ and roots in $K$.

Proposition

Suppose $g(x) \in k[x]$ is irreducible and there exists $\beta \in K$ such that $g(\beta) = 0$. Then $g( \sigma \beta) = 0 \quad \mbox{for all} \quad \sigma \in \mathrm{Aut}(K|k)$.

Proof

Suppose $g(x) = \sum_m b_m x^m$, then $\sigma b_m = b_m$, and hence $$0= \sigma g(\beta) = \sum_m \sigma b_m (\sigma \beta)^m = \sum_m b_m (\sigma \beta)^m = g(\sigma \beta). \qquad \square$$

One particularly important automorphism is complex conjugation. Suppose that $K | \mathbb Q$ is a number field, and $K \cong \mathbb Q(\alpha)$ for some non-real $\alpha$ (that is the minimal polynomial of $K$ has a non-real root in $\mathbb C$). Then, since complex conjugation is an automorphism of $\mathbb C | \mathbb R$, we have it is also an isomorphism on $K(\alpha) | \mathbb Q$. It follows that if $\beta \in \mathbb Q(\alpha)$ then $\overline \beta \in \mathbb Q(\alpha)$ as well and hence, as sets, $\mathbb Q(\alpha)=\mathbb Q(\overline \alpha)$, and algebraic operations in one of these embeddings can be found from the other by complex conjugation.

Let us distinguish the real and complex roots of $f(x)$ by setting $\alpha_1, \ldots, \alpha_{r_1}$ to be the real roots and $\beta_1, \overline{\beta_1}, \ldots, \beta_{r_2}, \overline{\beta_{r_2}}$ be the non-real complex roots. Clearly $r_1 + r_2 = d$. Then the embeddings $\mathbb Q(\alpha_1), \ldots, \mathbb Q(\alpha_{r_1}) , \mathbb Q(\beta_1) , \ldots, \mathbb Q(\beta_{r_2})$ are called the archimedean embeddings of $K$.

Splitting Fields

Galois theory is concerned about the zeros of rational polynomials and how their zeroes are permuted by the automorphisms of certain extensions of $\mathbb Q$ (which will come to be called Galois extensions). We already noted, that the automorphisms in $\mathrm{Aut}(K|k)$ preserve the set of zeroes in $K$ of any given polynomial $g(x) \in k[x]$. However, the construction $K$ makes no guarantee that a generic polynomial $g(x)$ will have a zero in $K$, and even for the minimal polynomial $f(x)$, the construction of $K$ only guarantees the existence of a single zero of $f(x)$.

In general, if we wanted an extension of $k$ that contains all the zeros of $f(x)$, we would first compute $K = k[x]/ f(x) \mathbb Q[x]$. $K$ contains at least one zero of $f(x)$, and if we factor it in $K[x]$ there will be a linear factor for each of those zeroes. We can then sequentially extend $K$ by constructing field extensions from the remaining irreducible factors of $f(x)$. Each time we extend fields by another irreducible factor, we add another zero of $f(x)$ to the resulting field extension. The process terminates after at most $d$ steps to produce the splitting field of $f(x)$. The splitting field of $f(x)$ has degree bounded by $d!$.

It is possible that the degree of the splitting field is as small as $d$, since it is possible, depending on the nature of $f(x)$, that $K[x]/f(x)\mathbb Q(x)$ itself contains $d$ zeros of $f(x)$.

Example

Suppose $p$ is prime and consider the $p$th cyclotomic polynomial $$\Phi_p(x) = x^{p-1} + x^{p-2} + \cdots + x + 1.$$ Suppose $\zeta$ is a zero of $\Phi_p(x)$ in $\mathbb Q[x]/\Phi_p(x) \mathbb Q[x]$, then it is easily verified that $\zeta^p = 1$.It follows that, if $\ell = 1, \ldots, p-1$, \begin{eqnarray}\Phi_p(\zeta^{\ell}) &=& \zeta^{\ell(p-1)} + \zeta^{\ell(p-2)} + \cdots + \zeta^{\ell} + 1 \\ &=& \zeta^{p-1} + \zeta^{p-2} + \cdots + \zeta + 1 \\ &=& 0.\end{eqnarray} It follows that $\zeta, \zeta^2, \ldots, \zeta^{p-1}$ are all $p-1$ roots of $\Phi_p(x)$ and hence $K = \mathbb Q[x]/\Phi_p(x)\mathbb Q[x]$ is the splitting field of $\Phi_p(x)$.

Galois Theory

Definition

If $K | k$ is a splitting field for a polynomial $g(x) \in k[x]$, then $K$ is said to be Galois over $k,$ and the group of automorphisms of $K$ which fix $k$ is called the Galois group and denoted $\mathrm{Gal}(K|k)$.

Claim

$K | k$ is Galois if and only if $\# \mathrm{Aut}(K|k) = [K : k]$.

We won’t prove this claim (though the only if direction is easy) because it is a bit fiddly with separability and involves a diversion into character theory. Some (most?) authors give this as the definition of Galois and prove that it implies the splitting field definition.

The main result in Galois Theory is a correspondence between intermediate fields of $K | k$ and subgroups of $\mathrm{Gal}(K|k)$. Let us write $G = \mathrm{Gal}(K|k)$ and suppose $H < G$ is a subgroup. Define $$K_H = \{ \gamma \in K : \sigma(\gamma) = \gamma \mbox{ for all } \sigma \in H \}.$$ It is easily verified that $K_H$ is a field, and $k \subset K_H \subset K$ (which we might abbreviate $K | K_H | k$). It will turn out that $H \leftrightarrow K_H$ will be a bijection (called the Galois correspondence) between subgroups of $G$ and intermediate fields of $K | k$.

This correspondence goes beyond a bijection, because there is an interpretation for $H$ and $G/H$ (as a subgroup in the case where $H$ is normal, but to some extent even as a set of cosets in the non-normal case) in terms of the groups of automorphisms $\mathrm{Gal}(K|K_H)$ and $\mathrm{Aut}(K_H|k)$. I hope you objected to the notational switch between $\mathrm{Gal}$ and $\mathrm{Aut}$ in the previous sentence, but it is correct. The fact that $K$ is a splitting field for a polynomial in $k[x]$ means that it is also the splitting field for a polynomial in $K_H[x]$ (namely any one of the irreducible factors of the original polynomial in $k[x]$) and hence $K | K_H$ is Galois and we use the notation $\mathrm{Gal}(K | K_H)$ for the group of automorphisms of $K$ preserving $K_H$. This is, unsurprisingly, equal to $H$. On the other hand, just because $K$ is the splitting field of a polynomial in $k[x]$ doesn’t imply that an intermediate field, such as $K_H$, must be a splitting field for that or any other polynomial in $k[x]$. Thus, in general we need to refer to the automorphism group of $K_H | k$ by $\mathrm{Aut}(K_H | k)$. It will turn out that when $H$ is normal in $G$ then $K_H | k$ is Galois, and $\mathrm{Gal}(K_H | k) \cong G/H$. This will all be enumerated in the Fundamental Theorem of Galois Theory, but we need to develop a few results first.

Given $\gamma \in K$, we call $\sigma \gamma; \sigma \in G$ the Galois conjugates of $\gamma$. Moreover, if $L$ is any intermediate field extension, $K | L | k$, then $\sigma$ gives an isomorphism from $L$ onto $\sigma(L)$ (which fixes $k$). In particular $K_H$ is isomorphic to its image $\sigma K_H$. Notice then that if $\psi \in \mathrm{Aut}(K_H | k)$, then $\sigma \psi \sigma^{-1}$ is an element of $\mathrm{Aut}(\sigma K_H | k)$.

Indeed, $\sigma H \sigma^{-1} = \mathrm{Aut}(\sigma K_H | k)$. We can make this more evocative by denoting the action by conjugation of $G$ on $H$ as $\sigma \cdot \psi = \sigma \psi \sigma^{-1}$, in which case, $$\sigma \cdot \mathrm{Aut}(K_H | k) = \mathrm{Aut}(\sigma K_H | k).$$ If $\sigma K_H = K_H$ for all $\sigma \in G$, then $GHG^{-1} = H$, that is $H$ is normal in $G$. On the other hand, if $H$ is normal in $G$, then $\sigma \psi \sigma^{-1} \in H$ and $\sigma K_H = K_H$ for all $\sigma \in G$.

Now, suppose $H$ is normal and $g(x) \in k[x]$ is a polynomial so that $K_H = k[x]/g(x)k[x]$. From the previous discussion, $x + g(x) k[x]$ is a zero of $g(x)$ in $K_H$, as are $\sigma (x + g(x) k[x])$ for all $\sigma \in G$. To establish $K_H | k$ is Galois, we need to show that the orbit of $x + g(x) k[x]$ under $G$ is equal to $[K_H : k]$. We know for $\sigma \in H$, $\sigma(x + g(x) k[x]) = x + g(x)k[x]$. On the other hand, if $\sigma(x + g(x) k[x]) = x + g(x)k[x]$ then $\sigma \in H$ because $\sigma$ is completely determined by its action on $x + g(x) k[x]$. Thus, the automorphisms of $\mathrm{Aut}(K_H|k)$ are in correspondence with $G/H$. We thus have $[K : K_H] = \#H$, $[K : k] = \#G$ and $[K_H : k] = \#G/H$. It follows that $\# \mathrm{Aut}(K_H|k) = [K_H : k]$ and $K_H | k$ is thus Galois.

To be sure, we have glossed over many details. However, many important observations are captured in the Fundamental Theorem of Galois Theory.

Fundamental Theorem of Galois Theory

Suppose $K | k$ is Galois and $G = \mathrm{Gal}(K|k)$.

CORRESPONDENCE

There is an inclusion reversing correspondence between intermediate fields of $K|k$ and subgroups of $H$.

Normality $\leftrightarrow$ Galois

$H$ is normal in $G$ if and only if $L|k$ is Galois. In this situation $\mathrm{Gal}(L|k) \cong G/H$.

The Correspondence Preserves Lattices

Suppose $H_1 \leftrightarrow L_1$ and $H_2 \leftrightarrow L_2$ for $H_1, H_2 \leq G$ and $L_1, L_2$ intermediate fields of $K|k$. Then $\langle H_1, H_2 \rangle \leftrightarrow L_1 \cap L_2$ and $H_1 \cap H_2 \leftrightarrow L_1 L_2$. (Here $\langle H_1, H_2 \rangle$ is the smallest subgroup of $G$ containing both $H_1$ and $H_2$ and $L_1 L_2$ is the smallest field containing both $L_1$ and $L_2$). Moreover the inclusions (e.g. $L_1 \cap L_2 \subset L_1 \subset L_1 L_2$) are reversed under the correspondence.

Here we write arrows for the inclusion map. The correspondence reverses inclusion.

The correspondence between subgroups of $\mathrm{Gal}(K|k)$ and subfields of $K|k$ is complete as the subfields of $\mathbb{Q}(i, \sqrt[8]{2})$ and subgroups of $G = \langle \sigma, \tau : \sigma^8 = \tau^2 = 1, \sigma \tau = \tau \sigma^3 \rangle$. This example was cribbed from Abstract Algebra, second edition by Dummit and Foote.

0

From Measures to Metrics on Pro-finite Completions

The complete 3-nary tree represents the family tree where each individual in a generation spawns (asexually) exactly three progeny in the subsequent generation. The image to the left represents 7 generations beginning from a single ancestor (the root) at the center of the image.

If we imagine the generations continuing ad infinitum, then we arrive at an object called the pro-finite completion of the tree. Loosely speaking this is the topological space which consists of all infinite paths from the root down through the (infinitely many) generations.

The pro-finite completion of the complete 3-nary tree can be put in correspondence with sequences of the form $(m_n)$ where each $m_n \in \{0,1,2\}$ as follows: Each descendent of an individual is labelled either 0, 1 or 2 (this can be done consistently by ordering, say, counterclockwise in the embedding above, but it doesn’t really matter so long as the labels are fixed for all time). A sequence starting, say, $(1,0,2,1,…)$ represents a child of (reading right to left) the first child of the second child of the zeroth child of the first child of the root. Admittedly ‘zeroth child’ sounds awkward, but we think of these as labels and not ordinals.

Visually, we may think of the pro-finite completion to be the boundary of the infinite graph, and the corresponding sequence $(m_n)$ as an address containing the information necessary to describe how to traverse the tree to get to that point on the boundary.

There are other embeddings of the complete 3-nary tree, including the ‘balloon embedding’ on the right. In this embedding the pro-finite completion is visualized as its (fractal) boundary. This embedding gives another construction of the fractal known as Serpienski’s Triangle.

In this embedding you may think of an ‘address’ of a point on the boundary as given by a sequence of ‘Left’, ‘Right’ and ‘Forward’ directions were you to drive to that point from the root along the edges of the graph. A bijection between $\{0,1,2\}$ and {Left, Right, Forward} will produce the sequence $(m_n)$.

Of course there’s nothing special about the 3-nary tree. We could start with any number of descendants per individual per generation. Indeed, we could let the number of descendants vary either between generations, or within a generation. We will see some examples of this soon.

Balloon embeddings of the complete 2-nary (binary), 4-nary and 5-nary trees. In each case the pro-finite completion is the fractal boundary of these graphs, and not the depicted edges and vertices.

Random Trees

There are lots of ways to make random graphs and trees, but here we will concentrate on a sort of random tree that will arise in the study of prime splitting in towers of number fields. We will suppose we start with a single ancestor (the root), and that each individual in the nth generation has an independent, identically distributed, bounded number of children. Note that the bound on the number of children may grow with generations, but for each generation there is some upper bound on the number of children an individual may have.

A simple example of a random tree where each individual has an equal chance of having 1, 2 or 3 offspring. The images are different embeddings of the same random tree.

Suppose the largest number of children an individual in the $n$the generation may have is $b_n$ (for instance, the random tree above has $b_n=3$ for all $n$). We call the sequence $(b_n)$ the sequence of generation bounds, and we call the tree where each individual in the $n$th generation has exactly $b_n$ children the complete $(b_n)$-nary tree. Every random tree with generation bounds $(b_n)$ can be embedded as a subtree in the complete $(b_n)$-nary tree.

As in the non-random case, the pro-finite completion of a random tree is the address, given as the directions necessary to traverse the tree from the root to a point on the `boundary’. Another way of representing the information given in the address is provided by the list of vertices $(v_n)$ one passes through on the voyage from the root. Here one assumes that the vertices are uniquely labelled. If $(v_n)$ is such a list of vertices we will write $v_m | v_n$ for all $m > n$. Loosely speaking, a vertex $v$ divides the vertex $w$ if $v$ is a vertex further down the tree from $w$. Put even more simply, $v | w$ if $v$ is descended from $w$. We will denote the root of the tree by $v_0$ and note that $v | v_0$ for all vertices $v$.

Let $B$ be the pro-finite completion of our (possibly random) tree as represented by sequences of vertices $(v_n)$, one per generation, with $v_m | v_n$ for all pairs $m > n$. For any vertex $w$ we define $$B(w) = \{ (v_n) \in B : w = v_m \mbox{ for some } m \}.$$ Loosely speaking $B(w)$ is the set of points in the pro-finite completion (boundary of the tree) that are downstream from vertex $w$.

$B(w)$ represents the part of the pro-finite completion that lies in the blue disk (left) or arc (right). Note that these are different representations of the same $B(w)$ on different embeddings of the same random tree.

Note that if $u$ and $w$ are different vertices, then either $B(u)$ and $B(w)$ are disjoint, or one is a subset of the other. It is worth supplying your own proof of this, or at least understanding why it is true from a picture.

$\sigma$-algebras and measures on $B$

We eventually want to talk about measures (and metrics) on the pro-finite completion of a random tree, but first we need a suitable $\sigma$-algebra. As usual, we actually define a nice collection of sets that we want to be in our $\sigma$-algebra and consider the smallest $\sigma$-algebra that does the trick. Dynkin’s $\pi$-$\lambda$ Theorem seems particularly salient here, and we define $\mathcal P$ to be the $\pi$-system given by all $B(w)$ for all vertices of our tree. That is $$\mathcal P = \{ B(v) : v \mbox{ is a vertex} \} \cup \emptyset.$$ We have to throw in the empty set, because a $\pi$-system is a collection of sets closed under intersection, and by our previous remarks, it is possible (common, in fact) for elements of $\mathcal P$ to be disjoint. We set $\mathcal D$ to be the $\sigma$-algebra on $B$ generated by $\mathcal P$. And we take $(B, \mathcal D)$ to be the measurable space in which all calculations occur.

Notice that, since the intersection of any two elements of $\mathcal P$ is again an element of $\mathcal P$ we see that elements of $\mathcal D$ are simply (possibly countable) disjoint unions of elements of $\mathcal P$. That is, for each set $A \in \mathcal D$ there is a (finite or) countable collection of vertices $V$, and a (finite or) countable $X \subset B$ such that $$A = \bigsqcup_{v \in V} B(v) \sqcup \bigsqcup_{x \in X} \{x \}.$$ The disjointness of this union implies we may do this in such a way that for any $u, v \in V$, $u \not | \;\; v$, and for any $x \in X$ and $v \in V$, $x \not \in B(v)$. We call $V$ a reduced set of vertices for $A$.

The $\pi$-$\lambda$ Theorem implies that a measure $\mu$ on $(B, \mathcal D)$ is determined completely by its values on $\mathcal P$. Note that if $w_1, \ldots, w_d$ are the child-vertices of vertex $w$, then $\mu(w) = \mu(B(w_1)) + \cdots + \mu(B(w_d))$ (and conversely, any collection of $\{m_v \in [0, \infty] : v \mbox{ a vertex}\}$ satisfying all consistency conditions of the form $m_w = m_{w_1} + \cdots + m_{w_d}$ will determine a measure on $(B, \mathcal D)$). There may be special measures on $(B, \mathcal D)$ depending on the construction of your tree, but for now we maintain complete generality, and see how various aspects of the measure interact to potentially give a metric on $B$.

Recall that an atom of the measure $\mu$ is a set $A \in \mathcal D$ such that $\mu(A) > 0$ and if $C \in \mathcal D$ is a proper subset of $A$ then $\mu(C) = 0$. By our construction of elements of $\mathcal D$ we see that if $A$ is an atom of $\mu$ then either $A = \{x \}$ for some $x \in B$, or $A = B(v)$ for some vertex $v.$ In fact, we will see that this latter situation is impossible. To see why, suppose the vertices $v_1, \ldots, v_d$ are the immediate descendants of $v$. Then, $$\mu(B(v)) = \mu(B(v_1)) + \cdots + \mu(B(v_d)).$$ If $d > 1$, It is not possible for $\mu(B(v)) > 0$ and $\mu(B(v_n)) = 0$ for all $n=1,\ldots, d$. Hence, if $v$ has more than one immediate descendent, $B(v)$ cannot be an atom of $\mu$. If, on the other hand $d = 1$ then $B(v) =B(v_1)$ and we can repeat our argument to show that either $B(v_1)$ is not an atom, or it only has one immediate descendant. It follows that if $B(v)$ is an atom then each descendent of $v$ has only one immediate descendent. That is, $B(v)$ contains only one $x \in B$, and hence if $B(v)$ is an atom, then in fact $B(v) = \{x \}$.

$B(v)$ can be a singleton only when all descendants of $v$ have only one immediate descendent. The only $B(v)$ that can be atoms of $\mu$ are singletons.

If $\mu$ has no atoms, then it is said to be diffuse. If $\mu(B(v)) > 0$ for all $B(v)$ that are not singletons, then we say $\mu$ is a full measure.

A pseudo-ultrametric formed from $\mu$

A metric on $B$ is a function $\delta: B \times B \rightarrow [0, \infty]$ such that for all $x,y,z \in B$,

  1. $\delta(x,x) = 0$
  2. $\delta(x, y) = 0$ implies $x = y$
  3. $\delta(x,y) = \delta(y,x)$
  4. $\delta(x,z) \leq \delta(x,y) + \delta(y,z)$

Note that we allow the possibility that $\delta$ is infinite. This is a slight generalization of the usual notion of a metric, but it disturbs very little. If we enforce the stronger requirement $$4′. \quad \delta(x,z) \leq \max\{\delta(x,y), \delta(y,z)\}$$ then we say $\delta$ is an ultrametric. If we instead, lose requirement (2), then we say $\delta$ is a pseudometric. Thus a pseudo-ultrametric $\delta$ satisfies for all $x,y,z \in B$,

  • $\delta(x,x) = 0$
  • $\delta(x,y) = \delta(y,x)$
  • $\delta(x,z) \leq \max\{ \delta(x,y), \delta(y,z) \}$

The third condition is called the ultrametric inequality or the strong triangle inequality.

Theorem

Given a measure $\mu$ on $(B, \mathcal D)$, define $\delta : B \times B \rightarrow [0,\infty]$ by $\delta(x,x) = 0$ for all $x \in B$, and for $x \neq y$ , $$\delta(x,y) = \inf\{ \mu(A) : A \in \mathcal P \mbox{ with } x, y \in A\}.$$ Then $\delta$ is a pseudo-ultrametric. Moreover, if $\mu$ is a full measure, then $\delta$ is an ultrametric.

Another way of defining $\delta$ is to first define the least common ancestor vertex of any two $x, y \in B$ by $a(x,y)$ in which case $\delta(x,y) = \mu(B(a(x,y))$. This definition only makes sense when $x \neq y$.

Proof

To show that $\delta$ is a pseudo-ultrametric, the only nontrivial condition to check is the ultrametric inequality. That is, for any $x, y, z \in B$, $\delta(x,z) \leq \max\{ \delta(x,y), \delta(y,z) \}$.

There are two cases. The first (slide up) we have $y \not \in B(a(x,z))$. In this case $B(a(x,z)) \subset B(a(y,z))$ and hence $\delta(x,z) \leq \delta(x,y)$. In the second case (slide down) $y \in B(a(x,z))$ and without loss of generality $B(a(x,z)) = B(a(y,z))$.

In the first case we get $B(a(x,z)) \subset B(a(y,z)) = B(a(x,y))$ and hence $\delta(x,z) \leq \max\{ \delta(x,y), \delta(y,z)\}$. (Note that equality is still possible in this case, because it is possible that $\mu(B(a(x,z)) = \mu(B(a(y,z))$ even when $B(a(x,z))$ is a proper subset of $B(a(y,z))$.

The second of these cases is a bit more delicate, but we see that, changing the labels if necessary, $a(x,z) = a(y,z)$. It follows that $\delta(x,z) = \delta(y,z)$ and $\delta(x,y) \leq \delta(y,z)$ which together yield $\delta(x,z) \leq \max\{\delta(x,y), \delta(y,z)\}$ as desired.

Notice that if $\mu$ is full, then $\delta(x,y) = \mu(B(a(x,y)) > 0$ and hence $\delta$ is in fact an ultrametric. $\square$

We say that $x \in B$ is isolated with respect to $\delta$ if there exists $\epsilon > 0$ such that $\delta(x,y) > \epsilon$ for all $y \neq x$. As the next result shows, isolated points come from atoms of $\mu$.

Lemma

Suppose $x \in B$ is such that $\{x\}$ is an atom of $\mu$. Then $x$ is isolated with respect to $\delta$. In particular, if $y \neq x$ then $\delta(x,y) \geq \mu\{x\}$.

Proof

By definition $x \in B(a(x,y))$. It follows that $\mu\{x\} \leq \mu(B(a(x,y)) = \delta(x,y)$. $\square$

0

Mandala Gardens and the Fibonacci Sequence, Maya Ward

Project Description

For my project I have created a mandala garden that is both functional and aesthetically pleasing according to mathematical principles, chiefly the Fibonacci sequence. The choice to use a mandala garden was a very intentional one. The mandala has often been regarded as one of the best representations of the intersection of math, aesthetics, and art. The mandala shown here, formed by octagonal shapes and concentric circles, helps represent some of the basic mathematical principles found in nature. The mandala garden also allows for maximum space to be used in an efficient manner, as it provides pathways to access most of the garden, so that all space can be utilized. The use of the keyholes as paths within the garden is both for aesthetic and functional purposes, it allows for the entirety of the garden to be utilized as it is now accessible by path. The keyhole paths, those that lay in-between the garden beds and are ended with either triangles or circles are a common feature seen in mandala gardens because they allow the gardener to access parts of the garden not within the circular beds. (Bittman 2016).

The first produce, found in the north, or top, circular bed has sets of the “three sisters” varieties growing. The three sisters is a type of strategy for planting that includes corn, bean, and squash. The three work together to help one another in the growing process. Corn stretches tall, which creates natural forming poles for the beans. The beans help the corn because the bean plants are nitrogen fixers, which helps corn which is a shallow rooted heavy feeder. The squash plants create living mulch which helps the heavy feeding corn and also the squash plant provides shade, killing weeds as well as warning off predators with spines. The three work in harmony, and while their harmonious relationship is intriguing for the mathematical nature in which they provide shade, surface area, and moisture for one another, they are also significant for culture. The three sisters are found in many native American communities and are seen as a gift from the great spirit to sustain life on earth  (“The Three Sisters”, 2014).

The second produce bed found on the south bottom quadrant circle bed has been divided into three sections. The sections are determined by how often the produce needs to be tended to, at the front is the produce most commonly harvested and in need of the most attention. This includes daily greens such as lettuce, kale, and chard/ the second third section has plants that are harvested slightly less, but still require somewhat frequent attention, this includes vegetables like eggplant and bell peppers. The furthest third included root crops which need little attention: ie. potatoes and other roots corps. While it may seem difficult to access this third quadrant, it can also be accessed from the keyhole paths found on either side of the garden bed. 

The sunflowers and lupins were chosen to line the garden bed because they are perennial nitrogen fixers. The nitrogen fixing plants work with the bacteria in the soil and ”capture the atmospheric nitrogen and convert it to bioavailable nitrates that the plants can use to grow” (Dana 2019). Sunflowers are especially fascinating because their pattern naturally represents the Fibonacci sequence. With most sunflowers, when you count the spirals on a sunflower both clockwise and counterclockwise you will pair of numbers from the Fibonacci sequence. Most commonly is 34 and 55 or 55 and 89 spirals, but some sunflowers have 89 and 144 spirals counterclockwise and clockwise (BohannonMay). The drawing of a sunflower included in my work shows how the spiral pattern of the internal section of the sunflower actually represents a graphing of the golden ration being x=[sqrt(5)-1]/2. This naturally forming pattern is one of the best examples of how integral the principles of mathematics are in the natural formation of plants.

The flower beds in the west and east circular beds have been very intentionally planted to represent the Fibonacci sequence. The different flowers planted starting the most NW circle on the left side to the most northern circle on the right start with 2 petals, then 3, 5, 8, 13, 21, 34, and 55. Furthermore the way in which I drew the garden to have circles that go in a specific pattern are meant to represent the pattern abstractly what the Fibonacci sequence looks like graphed. The flowers include, with petal numbers: begonia (2), trillium and iris (3), buttercup (5), columbine (5), larkspur (5), delphinium (8), black eyed Susan(13) Shasta daisy (21), aster (21), gaillardia (34), pyrethrum (34), and Michaelmas daisies (55) (Knot). Together, if planted in this pattern both represent the Fibonacci sequence in their physical layout and petal amounts.

The NW outermost side of the garden has another homage to the Fibonacci sequence. It is lined with the vines of the passionflower. The passionflower which does not want direct sun will do best here, and the passionflower has in order from most inner to outer 3, 5, 55, and two sets of 5 petals as the pattern, which can be seen in my drawing. This flower is special in the way that it, although out of order, represents three of the unique numbers found in Fibonacci sequence (Iannotti 2009). Furthermore, the ratio I have used to design where my walls of passionflower vines will be is a ration of 5:3:2:1, which is also found in the Fibonacci sequence.  On the SW side of the garden there are vines for peas, I have created this in the SE quadrant because the peas need large amounts of sun (Old Farmer’s Almanac). They also create symmetry in contrast to the passionflower vines opposite of them and follow the same 5:3:2:1 ratio. While initially these may be seen as just two walls of vines, their intentionality in both symmetry, ratio, and choice of plants helps further represent the Fibonnaci sequence within my garden.

In the rest of the sections outside of the beds not covered by lupins, sunflower, passionflower, or Pease I have intentionally planted spots of clovers and alfalfas. This is because clover and alfalfa are seen as living mulch. Living mulch is classified as “Any plant that is used to cover an area of soil and adds nutrients, enhances soil porosity, decreases weeds and prevents soil erosion, among other attributes” (Libretexts 2018). Alfalfa adds nitrogen and organic matter to the soil along with Phosphorus, Potassium, Calcium, Sulfur, Magnesium, Boron, Iron, Zinc. While it may not be interlaced with the vegetables and flowers, the soil can be used for planting of potter plants and spread within the beds. White clovers are widely adapted perennial nitrogen producers with that have sallow roots mass and tough stems, helping to protects form soil erosion. These nitrogen fixers are necessary for my garden because their relationship with soil bacteria converts N2 which plants cannot use to ammonia NH3 which plants are able to use to synthesize proteins (Libretexts 2018).

The pattern of the herb spiral is meant to mimic the logarithmic spiral. For consideration it is important to note that the south is the hottest, the east dries out earlier than the west and there is wetter soil at the bottom and driest at the top. You want to put your Mediterranean herbs/herbs that like to be dry in the top of the spiral and your mint and parsley as the bottom. At the top of the herb spiral is a succulent commonly known as hens and chickens (Sempervivum tectorum) which represents a beautiful naturally formed spiral.  Hens and chickens need well drained soil and full sun, perfect for the top of the spiral (Beaulieu 2005). The herbs in the garden starting from the top and to the bottom are calendula, mint, tarragon, parsley, chives, fennel, lemon balm, lavender, basil, chamomile, sage, oregano, rosemary, lemongrass with hens and chicken (“Gaia’s Garden, Hemenway 2009”; Engels 2015). All of these herbs along with the succulent are drawn in a diagram found alongside my garden design. 

Elements of symmetry, naturally forming Fibonacci sequences, use of mandalas and keyhole bed, and herb spirals all contribute to this garden. These elements not only allow for maximum efficiency of the garden, but also create aesthetically pleasing designs. While this garden may look like another garden, upon further inspection, the choices of plants, the uses of ratios, and the formation of spirals and patterns are all paying homage to the foundations of modern mathematics, the golden ratio, the Fibonacci sequence, and the logarithmic spiral. Alongside this, this garden is also meant to honor the naturally forming mathematically principles that people find so beautiful being pattern, symmetry, and bright colors. Not only is this garden both functional and intentionally created for the purposes of a multi-beneficiary ecosystem, is also made beautiful by a rooted foundation in mathematics. 

Works Consulted

Beaulieu, David. “Sempervivum: The ‘Always Live’ Plant.” The Spruce, The Spruce, 12 May 2020, www.thespruce.com/how-to-care-for-hens-and-chicks-plants-2132609.

Bittman, Erin. “Making Mandalas: the Ultimate Art-Math-Meditation Combo.” WeAreTeachers, 31 July 2017, www.weareteachers.com/making-mandalas-the-ultimate-art-math-meditation-combo/.

BohannonMay, John. “Sunflowers Show Complex Fibonacci Sequences.” Science, 9 Dec. 2017, www.sciencemag.org/news/2016/05/sunflowers-show-complex-fibonacci-sequences.

Dana, et al. “The Best Nitrogen Fixing Plants List for Your Garden.” Pīwakawaka Valley Homestead, 26 Apr. 2020, piwakawakavalley.co.nz/nitrogen-fixing-plants/.

Engels, Jonathon. “The Magic and Mystery of Constructing a Herb Spiral and Why Every Suburban Lawn Should Have One.” The Permaculture Research Institute, 17 Dec. 2016, www.permaculturenews.org/2015/04/17/the-magic-and-mystery-of-constructing-an-herb-spiral-and-why-every-suburban-lawn-should-have-one/.

Grant, Bonnie. “What Is Living Mulch: How To Use Living Mulch As A Ground Cover.” Gardening Know How, Gardening Know How, www.gardeningknowhow.com/edible/grains/cover-crops/living-mulch-ground-cover.htm.

“How to Grow Passion Flower.” How to Grow Passion Flower, The Sunday Gardner, www.sundaygardener.co.uk/howtogrowpassionflower.html.

Iannotti, Marie. “How to Easily Grow Exotic Passion Flowers.” The Spruce, The Spruce, 28 Sept. 2019, www.thespruce.com/passion-flowers-1403114.

Knott, R. “Fibonacci Numbers and Nature.” The Fibonacci Numbers and Golden Section in Nature – 1, www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#section4.

Libretexts. “31.3A: Nitrogen Fixation: Root and Bacteria Interactions.” Biology LibreTexts, Libretexts, 19 Nov. 2019, bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/Book%3A_General_Biology_(Boundless)/31%3A_Soil_and_Plant_Nutrition/31.3%3A__Nutritional_Adaptations_of_Plants/31.3A%3A_Nitrogen_Fixation%3A_Root_and_Bacteria_Interactions.

Nationwide, SARE. “White Clover.” SARE, www.sare.org/Learning-Center/Books/Managing-Cover-Crops-Profitably-3rd-Edition/Text-Version/Legume-Cover-Crops/White-Clover.

Old Farmer’s Almanac. “Peas.” Old Farmer’s Almanac, www.almanac.com/plant/peas.

Ratnala, Sujatha. “MATHEMATICS OF A FLOWER.” Medium, Medium, 4 Feb. 2018, medium.com/@sujatha.ratnala/mathematics-of-a-flower-e507ddaf0675.

Rodney. “Living Mulch: Benefits & Uses.” Traditional Gardening, 27 Jan. 2020, traditionalgardening.com/living-mulch/.

Schmidt, et al. “Alfalfa Living Mulch Advances Biological Control of Soybean Aphid.” OUP Academic, Oxford University Press, 1 Apr. 2007, academic.oup.com/ee/article/36/2/416/402457.

“The Three Sisters.” Grow Pittsburgh, 26 Nov. 2014, www.growpittsburgh.org/the-three-sisters/.

Toby, Hemenway. “Gaia’s Garden- A Guide to Home Scale Permaculture.” Permabox, Chelsea Green Publishing Company, 2009, permabox.ressources-permaculture.fr/1-PERMACULTURE/LIVRES/BOOKS_Permaculture-(english)/BOOK_Gaia-s-Garden—A-Guide-to-Home-Scale-Permaculture_by-Toby-Hemenway.pdf.

Waddington, Elizabeth. “How To Make An Herb Spiral.” Seed To Pantry School, 8 May 2020, seedtopantryschool.com/how-to-make-an-herb-spiral/.

0

untitled, Lauren Van Horn

Project Description

My creative project is focused on documenting how the sun travels across the sky and how lighting changes throughout the day. In order to do so, I took photos of the same seven locations at different times during the day. I took photos at dawn, in the afternoon, and at dusk. Furthermore, I wanted to ensure that each of the seven locations I chose to photograph offered a unique perspective. With this goal in mind, I chose a wide variety of locations to explore how sunlight is affected by different factors. There are four indoor locations and three outside ones. Three of the indoors locations have north-west facing windows. The last indoor location has a south-west facing window. The outside photos are taken facing south, south-east and east. To continue, taking the photos facing different directions allows a deeper insight to how the sun’s path interacts with earth.

Through the photos, I better understand what parts of my house receive sunlight and when. Some locations where I had thought there would be a large change in appearance only had a slight adjustment. There are certain factors that have influenced this outcome. For one, there were obstructions to the sunlight that I had not originally taken into account. Because of these interferences in sunlight, some parts of the house were never fully illuminated. As a result, some photos rely heavily on the hue of light in the photo to indicate the sun’s position. In regards to the sun’s light temperature, cloudy, overcast skies presents the bluest hues. On the other side of the spectrum, the warmest lighting occurs during sunrise and sunset. Right in the middle, neutral lighting is due to daylight at noon. In the photos, the viewer can notice the slight variations in lighting temperature as the day progresses, as well as between the sunny and cloudy day.

In a sense, the photos are acting as a modern, not very accurate, sundial. Sundials work by measuring the sun’s movement across the sky based off of shadows. By a similar manner, the way the light interacts with these locations can become an indicator of what time of day it is. This idea is most noticeable for the photos taken outside. Opposed to tracking the time of day due to the shadow cast by the sun, the viewer can track the time of day through how the patch of sun moves across the yard. Again, this is due to the obstructions of sunlight that were present in my backyard.

The path that the sun takes as it crosses the sky can be influenced by multiple factors. Throughout the year, the sun’s path varies slightly. The Earth has an axial tilt of around 23.5​° which affects the sun’s placement in the sky. ​Therefore, if I were to repeat this project during a different month but still photographing during the same hours, the locations would appear dramatically different. Another intriguing project would be to photograph the same location at the same time throughout the year to track the sun’s movement at varying times of the year. Furthermore, weather can influence the way light travels as well. Clouds disperse light, creating even lighting throughout the day. Because of these factors, it was important to photograph the locations both on a sunny day and on a cloudy day to see how the different lighting affects the image. Notably, there is less variation in both shadows and coloration in photos taken on the cloudy day than compared to photos taken on the sunny day.

During this process, there were a couple things that I noticed. First, capturing the subtle differences in lighting with a camera can be difficult. There are multiple factors that could have influenced this difficulty. The lack of variety in lighting on the cloudy day plays a significant role. Throughout the day, there was little need to change the camera settings. In comparison, the camera settings had to be altered almost every time a new photo was taken on the sunny day. When more light is introduced, the camera settings need to be more specific. The uniformity of the diffused lighting, on the other hand, did not need drastic camera alterations.

Overall, this project helps explain how lighting interacts with the appearance of a location. There are times of day and weather conditions that make photography easier and more “beautiful.” For consistent lighting and no harsh shadows, a cloudy day is perfect. Warm, sun-soaked images require a sunny day, preferably around golden hour. Lighting can affect the mood of photography in dramatic ways, as seen through this project.

0

The Mathematics of Irish Dance, Zoey Tevault

Artist Statement: The Mathematics of Irish Dance

In first grade, I was jumping around waiting for my teacher to check my work, and she told me I should look into Irish dance. The K-8 school had several dancers that competed internationally—like with actual Irish citizens—and some even won. While I never got past advanced beginner (because I didn’t like to go to competitions), Irish dance was so important to me in a lasting way.

Along with the dance lessons, we also learned about the Celtic culture we were practicing from Breda Yeates, an Irish immigrant. I’m a McElligott on my mom’s side which is very obviously an Irish name but I didn’t feel any connection to the culture in my own right; I never considered until now that my great-great-great-so on relatives might have practiced this dance that Yeates taught us. The dances developed when the Irish were enslaved and not allowed to celebrate their culture. The stiffness of the dancers made their movements unrecognizable as dance (or so I was told). The dancers’ hallmark look includes tight curls, a stiff paneled dress with elaborate embroidered decorations, and poodle socks. The Celtic knots are thought to have been popularized by the spread of Christianity but the designs that predate Christianity are hard to trace if they don’t depict an animal or recognizable symbol.

Celtic knotwork is often one continuous looping ribbon with distinct repeating patters. The circular and square knots are usually centered on a quadrant system and are exactly the same in each quadrant rotated. The unorthodox shaped ap knots are reflections over an axis. On the dresses, the skirt is paneled with the most elaborate design on the front of the dress. For the sake of design, most dresses use knots with 3 or more colorful ribbons. The adornments add a uniqueness to the stage. No two girls (from different schools) wear the same dress in the competitive stage.

For my creative project, I wanted to design a pattern that could be found on a typical dress. The way that Celtic knots curl on themselves and cross over and under in a specific pattern. By far, the hardest part of the design process was making the crossings evenly numbered; often, I would find myself at an impasse to determine what to do next because if I went one way I would put two “overs” or “unders” in a row which throws off the whole design. The design aspect is so intrinsically tied into mathematics, that there are computer programs specifically for designing Celtic knot patterns. Unfortunately, many links to the academic work with these programs are broken or inaccessible to the public but one free program, Knots3D is available on Windows. Unfortunately, I have a Mac and couldn’t run the program but from test pictures supplied on the download site, I was intrigued by the intricacy of the knots and their weaving. The nature of computer generation, of turning a 2D image into a 3D image, and furthermore, of a repeating loop image requires knowledge of topological spaces and geometry.

In my initial development of the knot, I focused too much on the drawing of the physical ribbon instead of controlling or containing the negative space. I was unsatisfied with the design but then, read up on the geometric patterns in Celtic knot work. Each knot has a “skeleton” with vertices in the negative spaces. Then, the midpoint of the side lengths is where the ribbon crosses over or under. The instructor describes the rule for crossing over or under akin to the right hand rule: if you were to physically pick up the ribbon to cross the lines, you would always cross left over right as you follow the path from am arbitrary point. The instructor pointed out that, ignorer to make a large knot, you first have to find a small knot pattern you like and tile it out through the entire area.

Regarding some basic knot theory from introductory documents from class, the definition of a knot is a (closed) tangled string. While the art form exists in a 2D place, the assumption for knots is that they are three dimensional. Without a loss of generality, we can view the 2D image as if it were 3D in order to apply the knot theory terms. Already, I have mentioned that Celtic knots are alternating knots. We can see that the classic Celtic trefoil is a knot and is the simplest nontrivial knot. At first glance, I would have assumed most Celtic knots are trivial but after revisiting some of the knot theory documents, I think that the knot in the outer ring of my design is nontrivial. There are sixteen figure-eight knots (thank you girl scouts) which would not pull apart. As for the inner design, I have no intuition about whether or not it would pull apart by looking at it alone. I presume that the majority of celtic knots are nontrivial because they develop so much upon themselves. The trefoil knot is the simplest knot topologically as mentioned previously, and the Celtic motif of the trefoil appears within much of the art that decorates dresses.

This project was a lot of fun; the nostalgia for me was so inspiring. I hope that in a few years, I can restart and join an academy. The class was asked what we would have changed about the project if we had chosen our topics at the end of the class instead of the beginning, and I will say that while I was inspired by the images in the knot theory papers we read for my topic, I would have decided to focus on the music and dance aspect of Irish culture. The difference between Reels, Slip Jigs, Light Jigs, Treble Jigss, and the Blackbird are so subtle but so profoundly tied into math. I think back to the only dance I remember in full:

O’er-2-3, o’er-2-3, o’er-2-3, o’er-2-3, hop-2-3-4-5-6-7, hop-step-step, o’er-2-3. Repeat. Hop- step-step, hop-step-step, hop-2-3-4-5-6-7, o’er-2-3-4-5-6-7, hop-step-step-and-a-point-hop-back. Repeat.

There absolutely has to be some fun mathematical connection in there!

0

Objects Warped by the Shape of Glass, Liza Richards

Project Description

For my creative project, I chose to demonstrate through photography how transparent glasses filled with water warp the appearance of the object behind them when looking through the glass. Differently shaped glass warps objects in a variety of ways. They may cause the appearance of the object to be smaller, larger, stretched, twisted, magnified, etc. The photograph I produced is designed to show exactly this and highlights their differences. While taking this photo I chose to use as plain of a background as I could find to emphasize the true focus of the photograph, the glasses and the distortions of the objects behind them. In order to make the ground more plain, I placed down pieces of blank white paper to avoid the patterns on the ground to take away from the flowers on display. I placed 7 flowers of different shapes, sizes, and colors behind the 4 glasses to use as the objects to be warped by the glass. My initial set up plan was to place each flower directly adjacent to one another, but I quickly figured out that with the ways the glasses warped the shapes, some of the flowers were not reflected through the glasses. To fix this, I had to move some flowers farther away from the others so that they would actually be seen through the glass which you can see with the photos I provided of the set up and the flowers without the glasses in front of them. I also struggled with the angles at which to take the photograph. I provided two somewhat different views of the glasses to allow the viewer to see how the angle at which you view the glasses changes the shape of the objects as well. You may notice that the the outermost right glass is the one that changed the appearance of the flowers the most with the different angle.

When comparing the photograph of the original flowers to the one of them behind the glasses, it can be observed that the flowers in the glass on the far left are pushed to the left side of the glass more and appear to have shrunken a little. This is due to the convex shape and curve of the glass. In the tallest glass, it is seen that there are actually two images of the same flower reflected in it. The one on the top is the same as the bottom except it has been flipped upside down. This is due to the concave curve of the glass near the top. In addition, the lower concave curve leads to the lower flower being shrunk. The glass in the middle right has multiple edges and corners, leading to multiple different refractions of the same flower through each section. The glass on the farthest right has an octagonal pattern engraved the glass while being concave and therefore magnifies and bends the flower into a combination of octagonal shapes.

We use glass in a multitude of ways similar to how they are used in this photograph. It is used in magnifying glasses, binoculars, and microscopes to allow the viewer to have a close up view of what they are observing. The way the curves of the glasses change the shapes of the flowers can also be seen in fun house mirrors, where the shape of the mirror warps the image reflected off of it. For these reasons, this photograph has mathematical connections to geometry, optics, and refraction.

Geometry is concerned with the properties of surfaces, solids, lines, etc. Geometry is an important part of mapping like navigation and astronomy. It can also be used in animations and designing. The geometry of the glasses changes how the object is viewed through the glass. The convex or concave curves will effect if an image is magnified or shrunk. The shapes in the glass lead to the object to be distorted to fit those same shapes as well. In addition, the appearance of the object through the glass depends on the original geometry of the object as well. Each of the different geometries present in these glasses changes the appearance of the object in their own unique way.

Optics is the study of sight and behavior of light and deflection. The geometry of the glasses deflects the original object, changing the appearance of the it in multiple different ways. The sole change in the appearance itself of the object through warped glass involves the study of optics. The study of optics is most known in our society through the creation of contact lenses and glasses to allow better eyesight. When we view the object through the water filled glasses, we are studying optics because we are observing how the image of the object changes as it is deflected through the glass.

Refraction is the phenomenon of light and objects being deflected in passing through an interface. It is mostly seen in rainbows, camera lenses, and binoculars, all involving how light is reflected and seen. When taking the photograph, I tried to avoid as much outside light as possible in order to allow only natural light to effect the appearance of the object. The deflection of the object can be considered a part of refraction. The natural light shining through the glasses allows the object to be warped in the way that it is. As the light passes through the glass, the portrayal of the object behind it changes.

In conclusion, as light shines through the water filled glasses, the shape of the glass effects the appearance of the flowers viewed through the transparent glass. The convex and concave shapes in the glasses shrink or magnify the flowers behind them. The edges and shapes in the glass lead to different distortions of the flowers through the glass. The changes in the appearance of the flowers are a result of the geometry of the flowers and the glasses, the optics and how the appearance of the flowers changes through the glass, and the refraction of light through the glass changing the shape of the flowers in the first place. A combination of these different types of mathematics is what explains the altered appearance of the flowers resting behind the glasses.

0

The Path of a Frisbee, Carmen Resnick

The project I will conduct will observe the path taken by a frisbee in flight from a “backhand huck.” There are two facets to my project, the aesthetics of the path taken and the math behind the aerodynamics of a frisbee. 

A backhand huck is used in ultimate frisbee at many of the most vital times in a game. An official game of ultimate frisbee begins with a “pull” where the frisbee is sent from one end of the field to the other by the defending team. Often, a backhand throw is used for this because it is the easiest to put enough power behind using the rotation of the torso and hips. To throw a backhand, the thrower steps with their dominant foot across their body and the disc starts at the non-dominant side. This movement will be demonstrated in the final project through a visual presentation. 

In a study done by Kathleen Baumback at the University of South Florida in 2010, the path of a frisbee was recorded in relationship to its initial angle. A series of calculations and formulas was used to understand the aerodynamics of a frisbee. Frisbees have multiple elements of their shape, as well as components such as spin, that enable them to maintain “lift” after the disc has left the hand of the thrower. This experiment was done to reproduce a similar study in 2005 which sought to “predict the path of a frisbee” (Baumback 3). This study recognizes the mathematic derivations to determine how the angle of initial release impacts the forces of lift and drag on the frisbee’s flight. All calculations and determinations were done through Java programming. 

The mathematics in my project are tailored more towards the aesthetics of a frisbee’s flight rather than the practicality of predicting it. I am interested also in taking the derivative of the determined graph (if possible) to understand the change in the frisbee’s angle over time. Two graphs will be created, one is similar to Baumback’s study of distance where the height reached by the frisbee will be observed over time and the other will study the angle of the disc with respect to the ground. 

Though not mathematically profound, this project serves to represent the beauty of seeing mathematical concepts in practice. Much more research could be done to comprehend how and why the physical shape and size of a frisbee impacts its flight. Additionally, the visualization of how a disc moves through the air will provide some clarity for me as a thrower to understand the disc’s momentum. 

There are many variables that influence the graphing of a frisbee in flight, both in the movement of the frisbee as well as the camera set-up and post-filming production content. I will video a frisbee in flight, ensuring that the camera is positioned behind the thrower. The angle behind the thrower will show the angle of the disc with relation to the ground, and the camera will be set up in a way that ensures that the bottom of the screen is parallel to the ground. From this angle, I will record both the degrees of inclination from the horizontal at various points in time, as well as the height of the disc from the ground. 

The original plan to observe the distance traveled by the disc was thrwarted when we took to the fields and realized that the disc’s distance would be impossible to capture in one frame of a video, and similarly difficult to measure on a screen. The height measurement is still accurate with regards to the screen because the camera was set up perfectly perpendicular to the ground.

Data and Methodology

Works Cited

Baumback, K. (2013). The Aerodynamics of Frisbee Flight. Undergraduate Journal of Mathematical Modeling: One + Two, 3(1). doi:10.5038/2326-3652.3.1.31

0