Is there such a norm on any totally disconnected local field?

Question

Let's set this definition of local field:

Let $\mathbb{K}$ be a field and a topological space (non-discrete and totally disconnected). Then $\mathbb{K}$ is called a local field if both $\mathbb{K}^+$ and $\mathbb{K}^*$ are locally compact Abelian groups, where $\mathbb{K}^+$ and $\mathbb{K}^*$ denote the additive and multiplicative groups of $\mathbb{K}$, respectively.

Is the following statement true?

Let $\mathbb{K}$ be any local field. Then there is an integer $q=p^r$, where $p$ is a fixed prime element of $\mathbb{K}$ and $r$ is a positive integer, and a norm $\lvert \,\cdot\, \rvert$ on $\mathbb{K}$ such that for all $x\in \mathbb{K}$ we have $\lvert x \rvert\geq 0$ and for each $x\in \mathbb{K}$ other than $0$ we get $\lvert x \rvert=q^k$ for some integer $k$.

Answer 1

A very nice answer to this question is provided in Section V.1 of the book "Symmetric Functions and Hall polynomials" by Ian Grant Macdonald. It is a beautiful argument, so I can't resist to give a sketch.

A local field $F$ is a non-discrete locally compact field. Therefore its underlying additive group has a Haar measure $\mu$. We define an absolute value $|\cdot|$ on $F$ by $|0| = 0$ and, for any nonzero element $a\in F$ and any measurable set $E\subset F$, $\mu(aE) = |a| \mu(E)$.

If $|\cdot|$ satisfies Archimedes' Axiom, then it is isomorphic to the field of real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$. Otherwise $F$ is totally disconnected. In this case $F$ is called a non-archimedean local field, and it is either a finite algebraic extension of the field $\mathbb{Q}_p$ of $p$-adic numbers (in characteristic 0), or a field of formal power series in one variable over a finite field (in characteristic $p$).

Let us assume that $F$ is a non-archimedean local field, which is your case. Define the "ring of integers" $O := \{a\in F: |a| \leq 1\}$ with maximal ideal $p := \{a\in F: |a| < 1\}$. Then $O$ is a complete discrete valuation ring with $F$ as field of fractions. Since $k := O/p$ is both compact and discrete, it is a finite field with, say, $q$ elements. Since any generator $\pi$ of $p$ satisfies $|\pi| = q^{-1}$, the absolute value $|a|$ of any nonzero $a\in F$ is some power of $q$ (this exponent is the normalized valuation of $a$).

Is there such a norm on any totally disconnected local field?. Available from: https://www.researchgate.net/post/Is_there_such_a_norm_on_any_totally_disconnected_local_field#5570b8cf5cd9e3a96a8b45c0 [accessed Jun 4, 2015].