Abstract
Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S-class numbers of intermediate fields, arising from Brauer–Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D2q and for arbitrary Galois-stable sets of primes S, containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D2q-extension from class numbers and S-class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.
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Schedule a callMore From: Journal für die reine und angewandte Mathematik (Crelles Journal)
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