Schur index (awaiting review)

Given a group $G$, a field $K \subseteq \mathbb{C}$ with $F$ a subfield of $K$, and a character $\chi$ of an irreducible $KG$-module $V$, define the field extension $F(\chi)$ of $F$ formed by adjoining to $F$ all the values the character takes on:

$$F(\chi)=F\left(\chi(g) : g \in G\right).$$

The Schur index of $\chi$ is the minimum of all indices $[E:F(\chi)]$ where E varies over all field $F(\chi) \subseteq E \subseteq K$ so that there is an $EG$-module $W$ with $V\cong K \otimes_E W$.