An <absolute value on a field> is a map $|\cdot|:K\to\mathbb R_{\geq0}$ satisfying $|x|=0\Leftrightarrow x=0$, $|xy|=|x||y|$, and $|x+y|\leq|x|+|y|$. It is <Non-Archimedean absolute value>[non-Archimedean] when the stronger inequality $|x+y|\leq\max(|x|,|y|)$ holds.

If $\operatorname{char}K=p>0$, then
$$
(x+y)^{p^r}=x^{p^r}+y^{p^r}.
$$
The ordinary triangle inequality gives
$$
|x+y|\leq2^{1/p^r}\max(|x|,|y|).
$$
Letting $r\to\infty$ proves the strong triangle inequality.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 1 / a / Solution

{title2=$|\cdot|$}
{wiki=Absolute_value_(algebra)}

An absolute value on a field is a nonnegative multiplicative function that vanishes only at zero and satisfies the triangle inequality.

= Absolute values on a field
{synonym}


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