The genus field and genus group in finite number fields

Extract

In this note those quotient groups of the absolute class group of number fields are to be studied which can be described in terms of absolutely Abelian fields. This investigation will be based on a suitable generalization of the classical concepts of the principal genus, the genus group and the genus field. One possible description of the genus group in a cyclic field is that as the maximal quotient group of the absolute class group which is characterized by rational congruence conditions, i.e. in terms of rational residue characters. From this point of view, however, the restriction to cyclic—or Abelian—fields is quite artificial; the given description can thus be taken as the definition of the genus group in any finite number field. In general the genus field will then no longer be absolutely Abelian; it can now be described as the maximal non-ramified extension obtained by composing the given field with absolutely Abelian fields.