Trivial absolute value
= Trivial absolute value
{title2=$|x|=\mathbf1_{x\ne0}$}
The trivial <absolute value on a field> is zero at zero and one elsewhere. Its metric <topology> is discrete. Its <valuation ring> and <residue field> are both the field itself. This is why rational absolute-value classification and nontrivial discrete-value compactness criteria need a nontriviality hypothesis.