Realizable classes of metacyclical extension to the lm degree

JOURNAL OF NUMBER THEORY

卷:130

Article

Sbeity, Farah¹  Sodaigui, Bouchaib¹ 

[1] Univ Valenciennes, Dept Math, F-59313 Valenciennes 9, France

关键词: Galois module structure;    Rings of integers;    Realizable classes;    Steinitz classes;    Maximal order;    Locally free class groups;    Frohlich-Lagrange resolvent;    Stickelberger ideal;   

DOI  :  10.1016/j.jnt.2009.10.007

来源: Elsevier

【 摘 要 】

Let k be a number field and O-k its ring of integers. Let l be a prime number and in a natural number. Let C (resp. H) be a cyclic group of order l (resp. in). Let Gamma =C x H be a metacyclic group of order fin, with H acting faithfully on C. Let M be a maximal. O-k-order in the semi-simple algebra K[Gamma] containing O-k[Gamma], and Cl(M) its locally free class group. We define the set R.(M) 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 Gamma, and for which [M circle times(Ok[Gamma]) O-N] = c, where ON is the ring of integers of N. In the present article, we define a subset of R.(M) and prove. by means of a description using a Stickelberger ideal, that it is a subgroup of Cl(M), under the hypothesis that k and the l-th cyclotomic field over Q are linearly disjoint. (C) 2009 Elsevier Inc. Tons droits reserves.

【 授权许可】

Free