P-Adic Numbers Introduction Cheat Sheet

Key Facts

• For a nonzero rational number $x$, the $p$-adic valuation $v_p(x)$ is the exponent of $p$ in the factorization of $x$, with $v_p(0)=\infty$.
• If $x=p^k\frac{a}{b}$ where $p\nmid a$ and $p\nmid b$, then $v_p(x)=k$.
• The $p$-adic absolute value is $|x|_p=p^{-v_p(x)}$ for $x\neq 0$, and $|0|_p=0$.
• The $p$-adic distance between $x$ and $y$ is $d_p(x,y)=|x-y|_p$.
• The ultrametric inequality says $|x+y|_p\leq \max(|x|_p,|y|_p)$, which is stronger than the usual triangle inequality.
• A sequence $(a_n)$ converges $p$-adically to $a$ exactly when $v_p(a_n-a)\to \infty$.
• Every $p$-adic integer has an expansion $a=a_0+a_1p+a_2p^2+\cdots$ with digits $a_i\in\{0,1,\dots,p-1\}$.
• The ring of $p$-adic integers is $\mathbb{Z}_p=\{x\in\mathbb{Q}_p:|x|_p\leq 1\}$.

Vocabulary

$p$-adic valuation

The function $v_p(x)$ that counts the exponent of the prime $p$ in a nonzero rational number $x$.

$p$-adic absolute value

The norm $|x|_p=p^{-v_p(x)}$ that makes numbers smaller when they are divisible by larger powers of $p$.

Ultrametric inequality

The rule $|x+y|_p\leq \max(|x|_p,|y|_p)$, which changes many geometric intuitions from ordinary real distance.

$p$-adic integer

An element of $\mathbb{Z}_p$ whose $p$-adic absolute value satisfies $|x|_p\leq 1$.

$p$-adic expansion

A series $a_0+a_1p+a_2p^2+\cdots$ with digits $a_i\in\{0,1,\dots,p-1\}$ that represents a $p$-adic integer.

$\mathbb{Q}_p$

The field of $p$-adic numbers, formed by completing $\mathbb{Q}$ using the $p$-adic distance.