On realizable Galois module classes and Steinitz classes of nonabelian extensions

JOURNAL OF NUMBER THEORY

卷:128

Article

Bruche, Clement¹  Sodaigui, Bouchaib¹ 

[1] Univ Valenciennes, Dept Math, F-59313 Valenciennes, France

关键词: galois module structure;    realizable classes;    steinitz classes;    maximal order;    froblich's hom-description of locally free class groups;    frohlich-lagrange resolvent;    embedding problem;    cyclic codes;    primitive polynomials;   

DOI  :  10.1016/j.jnt.2007.02.009

来源: Elsevier

【 摘 要 】

Let k be a number field and O-k its ring of integers. Let Gamma be a finite group, N/k a Galois extension with Galois group isomorphic to Gamma, and O-N the ring of integers of N. Let M be a maximal O-k-order in the semisimple algebra k[Gamma] containing O-k[Gamma], and Cl(M) its locally free class group. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to O-N the class of M circle times O-k[Gamma] O-N, denoted [M circle times(Ok[Gamma]) O-N], in Cl(M). We define the set 7Z(.A4) of realizable classes to be the set of classes c E Cl(M) such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to F, and for which [M circle times(Ok[Gamma]) O-N] = c. Let p be an odd prime number and let p be a primitive pth root of unity. In the present article, we prove, by means of a fairly explicit description, that R(M) is a subgroup of Cl(M) when xi(p)is an element of k and Gamma = V x(rho) C, where V is an F-p-vector space of dimension r >= 1, C a cyclic group of order p(r) - 1, and p a faithful representation of C in V; an example is the symmetric group S-3. In the proof, we use some properties of a cyclic code and solve an embedding problem connected with Steinitz classes. In addition, we determine the set of Steinitz classes of tame Galois extensions of k, with the above group as Galois group, and prove that it is a subgroup of the class group of k. (c) 2007 Elsevier Inc. All rights reserved.

【 授权许可】

Free