Relative Galois Module Structure and Steinitz Classes of Dihedral Extensions of Degree 8

✍ Scribed by Bouchaı̈b Sodaı̈gui

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 113 KB
• Volume: 223
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.8072

✦ Synopsis

Let k be a number field and O its ring of integers. Let ⌫ be the dihedral group k w x w x Ž . of order 8. Let M M be a maximal O -order in k ⌫ containing O ⌫ and C C l l M M k k Ž . its class group. We denote by R R M M the set of realizable classes, that is, the set of Ž . classes c g C C l l M M such that there exists a Galois extension Nrk at most tamely ramified, with Galois group isomorphic to ⌫ and the class of M M m O equal to O w ⌫ x N k Ž . c, where O is the ring of integers of N. In this article we prove that R R M M is a N Ž . subgroup of C C l l M M provided that k and the fourth cyclotomic field of ‫ޑ‬ are linearly disjoint, and the class number of k is odd. To this end we will solve an embedding problem connected with Steinitz classes of Galois extensions. In addition, for any k with odd class number, we show that the set of Steinitz classes of tame dihedral extension of k is the full class group of k.