Dyadic exercises for octahedral extensions II

This paper is the continuation of [P. Bayer, A. Rio, Dyadic exercices for octahedral extensions, J. Reine Angew. Math. 517 (1999) 1–17], where, by solving successive local embedding problems, we gave a description for all the Galois extensions of the dyadic field Q2 having as Galois group a subgroup of , the double cover of the symmetric group S4 with matrix model GL2(F3). The aim now is to obtain the complete chain of ramification groups for all of these extensions. As an application of this exhaustive local study we describe explicitly the arithmetic of the prime 2 in global S4 and extensions.