Given a finite group G and a number field k, a well-known conjecture asserts that the set Rt(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 Rt(k, G) when G is of odd order. More precisely, we define a subgroup W(k, G) of the class group of k and we prove Rt(k, G) ⊆ W(k, G). We show that equality holds for all groups of odd order for which a description of Rt(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 Rt(k, G) = W(k, G) when G is a group of order dividing 4 , where is an odd prime.
Abstract. We give a criterion for the vanishing of the Iwasawa λ-invariants of totally real number fields K based on the class number of K(ζ p ) by evaluating the p-adic L-functions at s = −1.
We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.
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