Steinitz classes of tamely ramified nonabelian extensions of odd prime power order

Abstract

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O k of k, which, together with the degree [K : k] of the extension determines the O k -module structure of O K . We call R t (k, G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group.In this paper we will develop some of the ideas contained in [8] to study some l-groups, where l is an odd prime number. In particular, together with [1] we will complete the study of realizable Steinitz classes for groups of order l 3 . We will also give an alternative proof of the results of [1], based on class field theory. R t (k, G) = {x ∈ Cl(k) : ∃K/k tame, Gal(K/k) ∼ = G, st(K/k) = x}.

An explicit candidate for the set of Steinitz classes of tame Galois extensions with fixed Galois group of odd order

“…The above result for n = 3 is actually proved in [1, Corollaire 1.2]. The following result slightly improves [9,Theorem 3.8].…”

“…Further, the same is true for all groups of order n and exponent n−1 , for any positive integer n (these are actually the groups studied in [9]). Lemma 5.1.…”

“…Let us assume that there exists K containing Q(ζ 7 + ζ −1 7 ), such that K/Q is tame and C(9)-Galois. Then 7 must be totally ramified in K/Q, since it is in Q(ζ 7 + ζ −1 7 )/Q, and so the inertia group must be the whole of C (9). By the proof of Proposition 2.9, this implies that 7 is totally split in Q(ζ 9 )/Q, which is not the case.…”

“…To show this, let us consider the following example. We take k = Q, k 1 = Q(ζ 7 + ζ −1 7 ), H = G = C(3) and G = C (9). Let us assume that there exists K containing Q(ζ 7 + ζ −1 7 ), such that K/Q is tame and C(9)-Galois.…”