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. It was noticed early on that the action of the Galois group puts constraints on the structure of the ideal class groups of normal extensions; most authors exploited only the action of cyclic subgroups of the Galois groups or restricted their attention to abelian extensions. See e.g. Inaba [10], Yokoyama [24], Iwasawa [12], Smith [19], Cornell & Rosen [5], and, more recently, Komatsu & Nakano [14]. Our aim is to describe a simple and general method that is applicable to arbitrary finite Galois groups. 1 Background from Algebraic Number Theory Let L be a number field, OL its ring of integers, IL its group of fractional ideals 6= (0), and Cl(L) the ideal class group of L in the usual (wide) sense. In this section we will collect some well known results and techniques that will be generalized subsequently. For relative extensions L/F of number fields, the relative norm NL/F : IL −→ IF induces a homomorphism Cl(L) −→ Cl(F ), which we will also denote by NL/F . The kernel Cl(L/F ) of this map is called the relative class group. The p-Sylow subgroup Clp(L/F ) of Cl(L/F ) is the kernel of the restriction of NL/F to the p-Sylow subgroup Clp(L) of Cl(L).