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The context for this paper is
the Inverse Galois Problem. First we give an if and only if condition that a finite
group is the group of a Galois regular extension of ℝ(X) with only real branch
points. It is that the group is generated by elements of order 2 (Theorem 1.1(a)). We
use previous work on the action of the complex conjugation on covers of
ℙ1. We also show each finite group is the Galois group of a Galois regular
extension of ℚtr(X). Here ℚtr is the field of all totally real algebraic numbers
(Theorem 5.7). Sections 1, 2, and 3 discuss consequences, generalizations, and
related questions.
The second part of the paper, §4 and §5, concerns descent of fields of definition
from ℝ to ℚ. Use of Hurwitz families reduces the problem to finding ℚ-rational point
on a special algebraic variety. Our first application considers realizing the symmetric
group Sm as the group of a Galois extension of ℚ(X), regular over ℚ, satisfying two
further conditions. These are that the extension has four branch points, and it also
has some totally real residue class field specializations. Such extensions exist for
m = 4, 5, 6, 7, 10 (Theorem 4.11).
Suppose that m is a prime larger than 7. Theorem 5.1 shows that the dihedral
group Dm of order 2m is not the group of a Galois regular extension of ℚ(X) with
fewer than 6 branch points. The proof interprets realization of certain dihedral
group covers as corresponding to rational points on modular curves. We
then apply Mazur’s Theorem. New results of Kamienny and Mazur suggest
that no bound on the number of branch points will allow realization of all
Dms.
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