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

Given a finite group G and a number field k, a well-known conjecture asserts that the set R-t(k, G) of Steinitz classes of tame G-Galois extensions of k is a subgroup of the ideal class group of k. In this paper, we investigate an explicit candidate for R-t(k, G) when G is of odd order. More precisely, we define a subgroup (k, G) of the class group of k and we prove R-t(k, G)subset of (k, G). We show that equality holds for all groups of odd order for which a description of R-t(k, G) is known so far. Furthermore, by refining techniques introduced by Cobbe [Steinitz classes of tamely ramified galois extensions of algebraic number fields, J. Number Theory 130 (2010) 1129-1154], we use the Shafarevich-Weil Theorem in cohomological class field theory, to construct some tame Galois extensions with a given Steinitz class. In particular, this allows us to prove the equality R-t(k, G)=(k, G) when G is a group of order dividing l(4), where l is an odd prime.