On finite linear groups stable under Galois operation

Abstract: We consider a Galois extension $E/F$ of characteristic 0 and realization fields of finite abelian subgroups $G\subset GL_n(E)$ of a given exponent $t$. We assume that $G$ is stable under the natural operation of the Galois group of $E/F$. It is proven that under some reasonable restrictions for $n$ any $E$ can be a realization field of $G$, while if all coefficients of matrices in $G$ are algebraic integers there are only finitely many fields $E$ of realization having a given degree $d$ for prescribed integers $n$ and $t$ or prescribed $n$ and $d$. Some related results and conjectures are considered.

Keywords: integral representations, Galois group, algebraic integers, Galois algebras

Classification (MSC2000): 20C10; 11R33

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