The $p$-adics and Related Concepts
Definition: An absolute value $| \cdot |$ on $\mathbb Q$ is a real valued function such that for any $x, y \in \mathbb Q$,
- Positive Definite: $|x| \geq 0$ and is equal to zero if and only if $x = 0$.
- Multiplicative: $| x y | = | x | \cdot | y |$
- Triangle Inequality: $|x + y| \leq |x| + |y|$
The usual absolute value, which we will denote here by $| \cdot |_{\infty}$ is sometimes called an archimedean absolute value because of the following.
Archimedes principal: For every integer $n$, there exists $x \in \mathbb Q$ such that $n – 1 < |x|_\infty \leq n$.
Another absolute value, the trivial absolute value, is given by $|0|_0 = 0$ and $|x|_0=1$ for all $x \neq 0$.
Given a rational number $x$ and a prime $p$ there exist integers $a, b, m$ such that $x = \pm p^m \frac{a}b$. We may assume that this is in lowest common terms, so in particular $a$ and $b$ are not divisible by $p$. We then define the $p$-adic absolute value by $|x|_p := p^{-m}$. Of course, just because we call something an absolute value doesn’t make it so, hence the next theorem.
Theorem: $| \cdot |_p$ is an absolute value.
The $p$-adic absolute value satisfies a stronger version of the triangle inequality.
Definition: If $| \cdot |$ is an absolute value on $\mathbb Q$ such that for any $x, y \in \mathbb Q$
- Strong Triangle Inequality: $|x + y| \leq \max\{|x|,|y|\}$,
Then $| \cdot |$ is called a non-archimedean absolute value.
The name non-archimedean is well placed since, if $n$ is an integer, and $| \cdot |$ non-archimedean, then repeated application of the strong triangle inequality implies $|n| \leq \max\{1, |n-1|\} \leq 1$.
Places and Ostrowski’s Theorem
Definition: Two absolute values $| \cdot |$ and $\| \cdot \|$ are equivalent if there exists positive real number $\alpha$ such that $| \cdot | = \| \cdot \|^{\alpha}$. An equivalence class of absolute values on $\mathbb Q$ is called a place of $\mathbb Q$.
Exercise: Show that if $p$ and $q$ are different primes, then $| \cdot |_p$ is not equivalent to $| \cdot |_q$.
The following theorem shows that we have pretty much discovered all the absolute values on $\mathbb Q$: there’s the trivial absolute value, there are those absolute values equivalent to $| \cdot |_{\infty}$ and there are those absolute values which are equivalent to $| \cdot |_p$ for some prime $p$. That’s it.
Ostrowski’s Theorem: Let $\| \cdot \|$ be a non-trivial absolute value on $\mathbb Q$. Then $\| \cdot \|$ is equivalent to $| \cdot |_p$ for exactly one $p \in \{\infty, 2, 3, 5, 7, \ldots \}$.
Another way of saying this is that if we call $\infty$ the infinite prime, then the set of places of $\mathbb Q$, $\mathcal M_{\mathbb Q}$, is in correspondence with the primes $P = \{\infty, 2, 3, 5, 7, \ldots \}$.
One interesting corollary of Ostrowski’s Theorem is that we can discover facts about the primes by understanding the absolutes values on $\mathbb Q$.