A natural question occurring in Galois theory is the following: Given a Galois realization of a factor group Ḡ of a group G, find a Galois realization of G compatible with that of Ḡ. This is called an embedding problem.

In Section 8.1 we study fields over which all embedding problems are solvable. This is a very strong property of a field. For a countable field, it determines the lattice of all separable algebraic extensions up to isomorphism. As shown in the last chapter, the field k(x) has this property, where k is algebraically closed of characteristic 0. Shafarevich's conjecture expects the same to hold for the cyclotomic field ℚab.

Actually it suffices to solve embedding problems with characteristic simple kernel. This leads to a basic dichotomy between the abelian and the nonabelian case. Section 8.2 leads up to the theorem that over a hilbertian field, each split abelian embedding problem is solvable. In Section 8.3 we discuss the notion of GAR-, GAL-, and GAP-realization, a way to deal with the case of centerless kernel. If each nonabelian finite simple group had a GAR-realization over ℚab, then this would imply Shafarevich's conjecture.

From Section 8.2 on we only consider fields of characteristic 0, our case of main interest. The case of positive characteristic can be included with the usual modifications.

Generalities

Fields over Which All Embedding Problems Are Solvable

See Section 7.5 for the definition of an embedding problem.