Genus theory and governing fields
Let \(L/K\) be a Galois extension of number fields. In this paper; the author develops an approach to the genus theory of \(L/K\) by introducing some governing field. When the restriction of each inertia group to the (local) abelianization is annihilated by a fixed prime number \(p\), this point of view allows the author to estimate the genus number of \(L/K\) with the aid of a subspace of the governing extension generated by some Frobenius elements. Then given a number field \(K\) and a possible genus number \(g\), the author derive information about the smallest prime ideals of \(K\) for which there exists a degree \(p\) cyclic extension \(L/K\) ramified only at these primes and having \(g\) as genus number.
scientific article; zbMATH DE number 6982028