Prime Ideals of a Galois Number Field and its Subfields

Abstract

A number field K which coincides with all its conjugate fields is called a Galois number field. If k is an arbitrary number field of degree m and k′,..., k (m−1) are the fields conjugate to k then a new field K can be composed from all the numbers of the fields k, k ′, ..., k (m−1); this field K is then a Galois number field which includes all the fields k, k ′,..., k (m−1) as subfields. Thus any arbitrary field k can always be thought of as a subfield of a Galois number field. It follows from this observation that in our investigation of the properties of algebraic numbers it will be no essential restriction to start with a Galois number field and then show how the factorisation laws for the ideals of such a field carry over to an arbitrary subfield.