Principal ideal theorems in the genus field for absolutely abelian extensions

This paper has the following contents. 1°. In an abelian extension field K over the rational number field, any ambiguous ideal is a principal ideal in the genus field in the wide sense. 2°. A number theoretical proof of the following. In a cyclic extension field K over the rational number field, any ambiguous class ideal is a principal ideal in the genus field in the wide sense.