The Schur index of an irreducible character

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. Proposition 4.3.2 in the cited notes of Zheng says that, as $V$ ranges over the irreducible $K$-representations, the absolutely irreducible constituents of $E{\otimes }_{K}V$ form a complete, nonrepeating list of the irreducible $E$-representations, grouped into Galois orbits. Consequently 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.