On Steinitz classes of nonabelian Ga lois extensions and p-ary cyclic Hamming codes.

Let k be a number field and Cl(k) its class group. Let Gamma be a finite group. Let R-t(k, Gamma) be the subset of Cl(k) consisting of those classes which are realizable as Steinitz classes of tamely ramified Galois extensions of k with Galois group isomorphic to Gamma. Let p be a prime number. In the present article, we suppose that Gamma = V x(rho) C, where V is an F5-vector space of dimension r 2, C a cyclic group of order (pr. - 1)/(p - 1) with gcd(r,p - 1) = 1, and p a faithful and irreducible 1F-representation of C in V. We prove that R-t(k,Gamma) is a subgroup of Cl(k) by means of an explicit description and properties of a p-ary cyclic Hamming code. 0 2013 Elsevier Inc. All rights reserved.