Brauer's cyclotomic criterion for splitting fields is recorded here only as an external theorem

Let $G$ be a finite group, and let $e$ be its exponent (The exponent of a finite group). Brauer's roots-of-unity criterion says that if a field $k$ contains a primitive $e$-th root of unity (The group ${\mu }_n(K)$ of $n$-th roots of unity in a field, and primitive $n$-th roots of unity, $t^n-1$ is separable over $K$ exactly when the characteristic does not divide $n$, and then a splitting field carries $n$ distinct $n$-th roots of unity), then $k$ is a splitting field for $G$ (A splitting field for a finite group: every irreducible representation has scalar endomorphism ring).

This theorem is recorded but not proved here.

The criterion is stronger than the algebraically closed-field consequence used later on this page, and it is genuinely external here. Its standard proof uses Brauer induction and the later character theory of finite groups, not only the group-ring dictionary and Schur's lemma developed on RT-1. The point of this remark is therefore negative: it blocks a tempting but unproved shortcut.