Dyadic exercises for octahedral extensions II Original Research Article

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 image having as Galois group a subgroup of image, the double cover of the symmetric group S4 with matrix model image. 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 image extensions.