Lemma 1

Tag 00LB, permanent. Cite it as [Bourbaki, Tag 00LB].

Algebra, VIII, p. 355, § 18. Simple Algebras over a Finite Field

Let L be a field, G a finite group, and $\chi$ a nontrivial homomorphism from G to the multiplicative group $L^*$ . We have $\sum_{x\in G}\chi (x) = 0$ .

By assumption, there exists an element $a$ of G such that $\chi (a)\not= 1$ ; since multiplication by $a$ is a permutation of G, we have

\sum_{x\in G}\chi (x) =\sum_{x\in G}\chi (ax) =\chi (a)\sum_{x\in G}\chi (x)

this gives Lemma 1.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Lemma 2, § 18

Cites

Nothing.