The Schur index of an irreducible character

Assume the Axiom of Choice (ZFC). Let $\chi$ be an irreducible complex character of a finite group $G$, put $K=\mathbb{Q}(\chi )$, and choose a finite cyclotomic splitting field $E\mathrm{/}K$; thus $K\subseteq E\subseteq \mathbb{C}$ and $E\mathrm{/}K$ is finite Galois. The finite-group specialization of Galois orbits classify simple modules after splitting base change applies to $K[G]$ and $E[G]$: it matches simple $K[G]$-modules bijectively with Galois orbits of simple $E[G]$-modules, and each orbit occurs in the extension of exactly one simple $K[G]$-module. Thus there is a unique irreducible $K$-representation $V$, up to isomorphism, whose scalar extension contains a representation affording $\chi$. In the decomposition from Scalar extension of an irreducible finite-group representation, write its common multiplicity as $m$. The Schur index of $\chi$ over $K$ is $m_K(\chi ):=m\mathrm{.}$ The next lemma proves that enlarging the chosen finite Galois splitting field does not change this integer; that is why this is a definition rather than an auxiliary choice.