Remark

Tag 0C4D, permanent. Cite it as [Bourbaki, Tag 0C4D].

General Topology, chapter IX, § 3. Metrizable groups, valued fields, normed spaces and algebras

If $x$ is a root of unity in a valued division ring, then $|x| = 1$ , for $x^n = 1$ implies $|x|^n = 1$ and hence $|x| = 1$ . In particular, the only absolute value on a finite field is the improper absolute value, since every element $\neq 0$ of such a field is a root of unity.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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