Genus theory, governing field, ramification and Frobenius

We develop, through a governing field, genus theory for a number field K with tame ramification in T and splitting in S, where T and S are finite disjoint sets of primes of K. This approach extends the one initiated by the second author in the case of the class group. We are able to express the S-T genus number of a cyclic extension L/K of degree p in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in L/K, in the Galois group of the underlying governing extension. For quadratic extensions L/Q, the matrices in question are constructed from the Legendre symbols of the primes ramified in L/Q and the primes of S.