Absolute value

Last updated January 27, 2022

Let $K$ be a field.

Definition. A map $|\cdot|\to\mathbb{R}_{\geq 0}$ is called an absolute value when

1. $|x|=0 \iff x=0$
2. $|\cdot|$ restricted to $K^\ast$ is a homomorphism $K^\ast\to\mathbb{R}^\ast_{\geq 0}$.
3. (Triangle inequality) $|x+y|\leq|x|+|y|$

Definition. An absolute value is called nonarchimedean if $$ |x+y|\leq\max(|x|,|y|). $$ Otherwise, it is called archimediean.

Equivalence

Definition. The absolute values $|\cdot|_1$ and $|\cdot|_2$ are called equivalent if either of the following holds:

1. $|\cdot|_2=|\cdot|_1^t$ for some $t>0$
2. They define the same topology on $K$.

Properties

1. If $|\cdot|$ is nonarchimedean and $x\in K$ can be expressed as a finite sum of distinct $x_i$, then $|x|=\max_{i}|x_i|$.
2. Consider the standard ring homomorphism $i:\mathbb{Z}\to K$. (i(\mathbb{Z}) is either $\mathbb{Z}$ or $\mathbb{F}_p$ depending on the characteristic of $K$.) Then $|\cdot|$ is nonarchimedian if and only if for all $x\in i(\mathbb{Z})$ we have $|x|\leq 1$.
   1. If $\text{char}(K)\neq 0$ then any absolute value on $K$ is nonarchimedean.

• Let $k=K(x)$, the rational polynomials in $K$. Suppose $|\cdot|$ is a nontrivial absolute value whose restriction to $K$ is trivial. Then it is nonarchimedean and either \begin{gathered}|x|=s^{-v_P(x)}\quad s\in\mathbb{R},s>1,P\in K[T]\text{ irreducible }, \text{deg}(P)\geq 1\text{, or} \ |x|=s^{\text{deg}(\text{numerator}(x))-\text{deg}(\text{denominator}(x))}\quad s\in\mathbb{R},s>1. \end{gathered}
  • These two are in fact the same, differing only by an automorphism of $K$.

Proof of (2). For the forward direction, each $x\in i(\mathbb{Z})$ can be written as a sum of $1$s. Since $|1|=1$, the non-archimedean property ensures $|x|\leq 1$.

Proof of (2.1) Every $x$ is such that $x^p=1$, so $|x^p|=1$ and since 1 is the only root of unity in $\mathbb{R}_{\geq 0}$ it follows $|x|=1$.

Over a field

1. Let $K$ be a field and $|\cdot|$ be an absolute value on it. Then
   1. Addition, subtraction, and multiplication, and absolute value are continuous functions.
   2. The absolute value may be extended to the completion of $K$, which is also a fieldd. It becomes an absolute value on the completion. If the original absolute value was nonarchimedean, so is the extended one.
   3. If the absolute value is nonarchimedean, then the set of absolute values of $K$ is the same as the set of absolute values of the completion of $K$.
2. Let $|\cdot|$ be a nonarchimedean absolute value on $K$. Suppose $K$ is complete with respect to this absolute value. Then $$ \sum_{i=1}^\infty x_i,\quad x_i\in K $$ converges if and only if $\lim_{i\to\infty}x_i=0$.
3. Let $K/k$ be a finite field extension, with absolute value $|\cdot|$ whose restriction to $k$ gives $k$ the structure of a complete metric space. Then the topology of the metric space $K$ coincides with the topology of $k^n$, i.e. the absolute value is entirely defined by its restriction to $k$.

Examples

• The trivial absolute value has $|0|=0$ and $|x|=1$ otherwise.