Nigel P Byott, Cornelius Greither, Bouchab Sodagui
Classes réalisables d'extensions non abéliennes
Let k be a number field and Ok its ring of integers. Let &Ggr; be a finite group, N/k a Galois extension with Galois group isomorphic to &Ggr;, and ON the ring of integers of N. Let ℳ be a maximal Ok-order in the semi-simple algebra k[&Ggr;] containing Ok[&Ggr;], and Cl(ℳ) its locally free classgroup. When N/k is tame (i.e., at most tamely ramified), extension of scalars allows us to assign to ON the class of , denoted , in Cl(ℳ). We define the set ℛ(ℳ) of realizable classes to be the set of classes c ∈ Cl(ℳ) such that there exists a Galois extension N/k which is tame, with Galois group isomorphic to &Ggr;, and for which . In the present article, we prove, by means of a fairly explicit description, that ℛ(ℳ) is a subgroup of Cl(ℳ) when , where V is an 𝔽2-vector space of dimension r ≧ 2, C a cyclic group of order 2r − 1, and p a faithful representation of C in V; an example is the alternating group A4. In the proof, we use some properties of the binary Hamming 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 classgroup of k.
Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter
Print ISSN: 0075-4102
Volume: 2006, 12/2006
Pages: 1 - 27
Show full article (external site)
Show all available items of this journal
Search
