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7 - The Descent from ℂ to k̄
Published online by Cambridge University Press: 22 September 2009
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 = Hn,s with generators h1,…, hs satisfying:
1. For any group G of order ≤ n, and g1,…, gs ∈ G, there is a homomorphism H → G sending hi to gi (for i = 1,…, s).
2. The intersection of all normal subgroups of H of index ≤ n is trivial.
3. If h1′,…, hs′ are generators of a group H′ satisfying (2) then there is a surjective homomorphism H → H′ sending hi to hi′.
4. If h1′,…, hs′ are any s elements generating H then there is an automorphism of H sending hi to hi′.
Proof. Let ℱs be the free group on s generators. First note that ℱs has only finitely many normal subgroups of index ≤ n.
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- Groups as Galois GroupsAn Introduction, pp. 119 - 129Publisher: Cambridge University PressPrint publication year: 1996