The purpose of this note is to record for my algebra class a proof of the Inverse Galois Problem for finite abelian groups. Recall that the Inverse Galois Problem is stated as follows: Given a finite group G, is there a Galois extension Q ⊆ K such Gal(K/Q) = G? The crucial point in the problem is that the base field is Q, since given any finite group G, there is a Galois extension of fields F ⊆...

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

The goal of this paper is to provide background, heuristics and motivation for several conjectures of Deligne [12, 8.2, p. 163], [12, 8.9.5, p. 168] and [23, p. 300] and Goncharov [15, Conj. 2.1], presumably along the lines used to arrive at them. A complete proof of the third and fourth of these conjectures and partial solutions of the first two are given in [19]. A second goal of this paper i...