On Steinitz classes of nonabelian Galois extensions and p-ary cyclic Hamming codes

Let k be a number field and Cl(k) its class group. Let Γ be a finite group. Let Rt(k,Γ) 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 Γ. Let p be a prime number. In the present article, we suppose that Γ=V⋊ρC, where V is an Fp-vector space of dimension r⩾2, C a cyclic group of order (pr−1)/(p−1) with gcd(r,p−1)=1, and ρ a faithful and irreducible Fp-representation of C in V. We prove that Rt(k,Γ) is a subgroup of Cl(k) by means of an explicit description and properties of a p-ary cyclic Hamming code.