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Let  F / k  be a Galois extension of number fields with dihedral Galois group of order 2 q , 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  D  2 q   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  D  2 q  -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.

On Brauer–Kuroda type relations of S -class numbers in dihedral extensions