I want to ask about realization problem of characters of a finite group. In this question, $G$, $m$, $\chi$, $K$, and $L$ are defined as follows:
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$G$: a finite group
$m$: the exponent of $G$
$\chi$: a faithful irreducible complex character of $G$
$K$, $L$: number fields where $\chi$ is realizable
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I wonder if $\chi$ is realizable over $K\cap L$. More precisely, you may assume that $L=\mathbb{Q}\left(\zeta_{m}\right)$ because Brauer's theorem says that $\chi$ realizes over $\mathbb{Q}\left(\zeta_{m}\right)$.
If you need, you may assume some conditions on $G$.