p-Adic Absolute Value

Definition A field norm, denoted by $|\cdot|$, on the field $F$ is a function from $F$ to the set of nonnegative numbers in an ordered field $R$ such that (1) $|x|=0$ if and only if $x=0$; (2) $|xy|=|x||y|$; (3) $|x+y|\leq|x|+|y|$.

Remark The ordered field is usually chosen to be the field of reals $\mathbb{R}$. However, to construct of $\mathbb{R}$ itself, the ordered field is the field of rationals $\mathbb{Q}$.

For example, the trivial norm on F is a norm $|\cdot|$ such that $|0|=0$ and $|x|=1$ for $x\neq 0$.

** Alternative Norms on Rationals** For any prime number $p$ and nonzero rational $a$, let $v_p(a)$ be the p-adic valuation of $a$. Define a map $|\cdot |_p$ on $\mathbb{Q}$ as follows: $$|x|_p=\begin{cases}p^{-v_p(x)} & \text{if }x\neq 0; \\ 0 & \text{if } x=0.\end{cases}$$

Proposition $|\cdot|_p$ is a norm on $\mathbb{Q}$. We call it the $p$-adic absolute value (or $p$-adic norm) on $\mathbb{Q}$.

Try prove it yourself using definition of a (field) norm before we continue.

Hint: You will need $v_p(a+b) \ge \min(v_p(a),v_p(b))$ for integer $a,b$. (Essentially, this is a practise problem here.)

Proof (sketch) We only check the triangle inequality. Notice that Therefore, we obtain $$|x+y|_p\le \max(|x|_p,|y|_p)\le |x|_p+|y|_p.$$ QED

A norm on the field $F$ naturally gives rise to a metric $d(x,y)=|x-y|$ on $F$. For example, the standard metric on the rationals is defined by the absolute value, namely, $d(x,y)=|x-y|$, where $x,y\in \mathbb{Q}$, and $|\cdot|$ is the standard norm, i.e., the absolute value, on $\mathbb{Q}$.

It would be great if we can classify all nontrivial norms modulo the norm equivalence. The following theorem due to Ostrowski classifies all possible norms on the rationals up to norm equivalence. We denote the standard absolute value by $|\cdot|_\infty$.

Definition Two norms on $F$, $|\cdot|_1,|\cdot|_2$, are equivalent if and only if for every sequence $(a_n)$, it is a Cauchy sequence with respect to $d_1$ if and only if it is so with respect to $d_2$, where $d_1,d_2$ are the metrics determined by $|\cdot|_1,|\cdot|_2$ respectively.

Theorem (OSTROWSKI 1916) Every nontrivial norm $|\cdot|$ on $\mathbb{Q}$ is equivalent to $|\cdot|_p$ for some prime $p$ or for $p=\infty$.

As one might have noticed, the $p$-adic norm satisfies an inequality stronger than the triangle inequality, namely $|a\pm b|_p\le \max(|a|_p,|b|_p)$. Such norm satisfying the stronger triangle inequalty is called non-Archimedean.

The world of non-Archimedean norm is good and weird. Here are two testimonies.

Proposition (No more scalene triangles!) If $|x|\neq |y|$, then $|x\pm y|=\max(|x|,|y|)$.

Proposition (All points are centers!) $D(a,r^-)=D(b,r^-)$ for all $b \in D(a,r^-)$ and $D(a,r)=D(b,r)$ for all $b\in D(a,r)$, where $D(c,r^-)=\{x:|x-c|<r\}$ and $D(c,r)=\{x:|x-c|\le r\}$.

Let $x$ be a rational number and $p$ be a prime. $x = \dfrac{p^ar}{s}$ for some integer $a$ and integers $r$ and $s$ not divisible by p. Then the p-adic norm of x is $|x|_{p} = p^{-a}$. $|0|_{p} = 0$.

For example, consider $x = \dfrac{96}{101}$. Since $96 = 2^5\cdot3$, $|x|_{2} = 2^{-5}$.

Source: http://mathworld.wolfram.com/p-adicNumber.html