In this note we develop an approach to genus theory for a Galois extension L/K of number fields 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 us 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, we 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.

The author was partially supported by the ANR project FLAIR (ANR-17-CE40-0012). This work has been done during a visit at Harbin Institute of Technology. The author thanks the Institute for Advanced in Mathematics of HIT for providing a beautiful research atmosphere.