Galois action on class groups

It is well known that the Galois group of an extension L/F puts constraints on the structure of the relative ideal class group Cl(L/F). Explicit results, however, hardly ever go beyond the semisimple abelian case, where L/F is abelian (in general cyclic) and where (L:F) and #Cl(L/F) are coprime. Using only basic parts of the theory of group representations, we give a unified approach to these as well as more general results.