Construction of genus field and some applications

Let k be a number field of finite degree. The narrow genus field K of k (genus field of k in the sense of Fröhlich) is defined as the maximal extension of k which is unramified at all finite primes of k of the form kk₁, where k₁ is an Abelian number field. In this article, K is determined and some applications are given. The results indicate a possibility that many class field theoretic properties of normal number fields could be extended to nonnormal number fields.