7 - The Descent from ℂ to k̄

Summary

In this chapter we prove that a finite extension of ℂ(x) can be described by an equation whose coefficients are algebraic over the field generated by the branch points. Thus the algebraic version of RET (Theorem 2.13) remains true if ℂ is replaced by any algebraically closed subfield k. The most interesting case is of course. Applications include the solvability of all embedding problems over k(x), as well as the existence of a unique minimal field of definition for each FG-extension of ℂ(x) whose Galois group has trivial center.

Extensions of ℂ(x) Unramified Outside a Given Finite Set

Lemma 7.1 Let s, n ∈ ℕ. There exists a finite group H = H_n,s with generators h₁,…, h_s satisfying:

1. 1. For any group G of order ≤ n, and g₁,…, g_s ∈ G, there is a homomorphism H → G sending h_i to g_i (for i = 1,…, s).

2. 2. The intersection of all normal subgroups of H of index ≤ n is trivial.

3. 3. If h₁′,…, h_s′ are generators of a group H′ satisfying (2) then there is a surjective homomorphism H → H′ sending h_i to h_i′.

4. 4. If h₁′,…, h_s′ are any s elements generating H then there is an automorphism of H sending h_i to h_i′.

Proof. Let ℱ_s be the free group on s generators. First note that ℱ_s has only finitely many normal subgroups of index ≤ n.