We fix following objects:

(1) $G$ is a finite group.

(2) $\chi$ is complex irreducible character of $G$.

(3) $m$ is the Schur index of $\chi$ w.r.t. the rational field $\mathbb{Q}$.

(4) All the fields appearing below will be of characteristic $0$.

The Schur index of $\chi$ is $m$ w.r.t. $\mathbb{Q}$; whereas it is $1$ w.r.t. any splitting field $F$ of $\chi$. Thus, it is natural to ask the following question.

Q.Given a divisor $m'$ of $m$, can $m'$ occur as Schur index of $\chi$ w.r.t. some extension of $\mathbb{Q}$ in $\mathbb{C}$?

Such question may have been considered in different contexts because the Schur index of an absolutely irreducible character w.r.t. $F$ is also the Schur index of simple component of $F[G]$ corresponding to $\chi$ (in the sense - $\chi$ appears as an irreducible component of an $F$-irreducible charatcer when scalars are extended from $F$ to $\mathbb{C}$). The simple components of $F[G]$ are particular examples of simple algebras, where division algebras are important and Schur index is also defined for division algebras. Thus, it may be the case that the problem of above kind could have been considered in the context of *index of simple algebras*. But I do not have idea. Can one suggest some way towards solution to above question, for example, literature if it is considered.