Genus theory, governing field, ramification and Frobenius

We develop, through a governing field, genus theory for a number field ${\rm K}$ with tame ramification in $T$ and splitting in $S$, where $T$ and $S$ are finite disjoint sets of primes of ${\rm K}$. 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 ${\rm L}/{\rm K}$ of degree $p$ in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in ${\rm L}/{\rm K}$, in the Galois group of the underlying governing extension. For quadratic extensions ${\rm L}/\mathbb Q$, the matrices in question are constructed from the Legendre symbols of the primes ramified in ${\rm L}/\mathbb Q$ and the primes of $S$.