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

Keywords:

class group; representation
Full Text:

References:

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