Dyadic exercises for octahedral extensions - In memory of Jurgen Neukirch

The object of this paper is the description of a family of extensions of oc tahedral type of the field Q(2) of the 2-adic numbers. Let (S) over tilde(4 ) be the double cover of the symmetric group S-4 that has as matrix model t he general linear group GL(2, F-3). Module Q(2)-isomorphisms, the number of Galois extensions of Q(2) having as Galois group a nontrivial subgroup of (S) over tilde(4) totals 130. Eight of them were given by Well in his paper [W]. We present a complete list of all these extensions by determining, in each case, an irreducible equation and the discriminant of the field. The method we use mainly consists in the resolution of successive local Galois embedding problems with kernel of order two.