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Galois action on class groups. (English) Zbl 1032.12008

The goal of this article is to give a unified approach for the study of the constraints placed on the structure of the class group of a normal extension by the action of the Galois group of the extension. For a number field \(K\) and a prime \(p\), let \(\text{Cl}(K)\) and \(\text{CL}_p(K)\) denote the class group of \(K\) and its \(p\)-Sylow subgroup. For a normal extension \(L/F\) of number fields the relative class group is the kernel of the norm mapping \(N_{L/F}: \text{Cl}(L)\to \text{Cl}(F)\). It and its \(p\)-Sylow subgroup will be denoted by \(C(L/F)\) and \(\text{CL}_p(L/F)\). It is first shown that if \(p\) does not divide \([L:F]\) then \(\text{Cl}_p(L)\cong \text{CL}_p(L/F)\times \text{Cl}_p(F)\).
The main theorem of the article states: Let \(p\) be a prime not dividing the order of the Galois group, \(\Gamma= \text{Gal}(L/F)\) and assume that \(\text{Cl}(K/F)= 1\) for all normal extensions \(K/F\) with \(F\subseteq K\subset L\), but \(K\neq L\). Let \(\Phi\) denote the representation \(\Phi: \Gamma\to \operatorname{Aut}(C)\) induced by the action of \(\Gamma\) on \(C= \text{Cl}(L/F)/\text{Cl}(L/F)^p\); if \(C\neq 1\), then \(\Phi\) is faithful. If the degrees of all irreducible faithful \(F_p\)-characters \(\chi\) of \(T\) are divisible by \(f\) then the rank \(\text{Cl}_p(L)\equiv 0\pmod f\). If, in addition, \([F_p(\chi): F_p]= r\) for all these \(\chi\), then the rank \(\text{Cl}(L)_p\equiv 0\pmod{rf}\). Here \(F_p\) denotes the field of \(p\) elements and \(F_p(\chi)\) denotes the smallest extension field of \(F_p\) containing all values in the image of \(\chi\).
Applications of the results are given when \(\Gamma\) is the Klein 4-group, the quaternion and dihedral groups of order 8 and \(A_4\).

MSC:

12F20 Transcendental field extensions
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R29 Class numbers, class groups, discriminants
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