It is well known that the Galois group of an extension puts constraints on the structure of the relative ideal class groups. Using only basic parts of the theory of group representations, we give a unified approach to such results.
Belyi-extending maps and the Galois action on dessins d'enfants
On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group
Galois action on the $\bar\mathbb{Q}$-isogeny classes of abelian surfaces with quaternionic multiplication
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