Normal Extensions of Quartic Fields with the Symmetric Group

Abstract

In an earlier2 paper of the author, the Hilbert subgroup series in a normal algebraic number field N of degree 24 whose group is S4, the symmetric group of degree four, were studied. The study of those subgroup series gave rise to a study of the characterization of those fields N = B(VII, V,.s2), where B is normal of degree six with group S3, the symmetric group of degree three, ju and MA2 are quantities in B, and N is the field described above. In the present paper necessary and sufficient conditions are given that N = B(VII, VM2) have the prescribed properties. In ?4 an interesting result is adduced concerning the extension of totalimaginary normal fields B with group S3 to normal fields N with group S4. In the final section the question of the realization of the various entries in the tables3 of our earlier paper, is considered and it is shown that many of those cases are realizable. It must be mentioned, finally, that in the entire paper, with the exception of ?4, the reference field may be taken to be an arbitrary algebraic number field instead of R, the field of the rational numbers, as we do.