Arithmetic of octahedral sextics

Abstract

Given a quartic field with S4 Galois group, we relate its ramification to that of the non-Galois sextic subfields of its Galois closure, and we construct explicit generators of these sextic fields from that of the quartic field, and vice versa. This allows us to recover examples of S4-sextic fields of Cohen and of Tate unramified outside 229, and to easily determine the tame part of the conductor of an octahedral Artin representation. We study class number divisibility arising from S4-quartics whose discriminants are odd and square-free, we explicitly construct infinitely many S4-quartics whose discriminants are −1 times a square, and experimental data suggest two surprising conjectures about S4-quartic fields over Q unramified outside one finite prime.