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 $\mathbb Z_p$.

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