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 $x = 0$.*Multiplicativ*e: $| x y | = | x | \cdot | y |$*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 $\mathbb Q$.