Nonrigid constructions in Galois theory
This paper considers diophantine problems with realizing a finite group \(G\) as the Galois group of a regular extension \(L/\mathbb{Q}(x)\) of \(\mathbb{Q}\): \(\overline{\mathbb{Q}}\cap L= \mathbb{Q}\). It illustrates, for example, why you can't dictate where to put the branch points (ramified places) of \(L/\mathbb{Q}(x)\). There is an if and only if condition that \(G\) is the group of a Galois regular extension of \(\mathbb{R}(x)\) with only real branch points. It is that involutions generate \(G\) (Theorem 1.1 (a)). If you can't realize \(G\) with real branch points, you certainly can't realize it with rational branch points. A more sophisticated topic produces a group theory test for finding \(L/\mathbb{R}(x)\) with real branch points, regular but not necessarily Galois, with Galois closure over \(\mathbb{C}\) the (geometric) group \(G\). This direction concludes with a profinite topic. Given \(G\) passing this test, is there a totally nonsplit extension of \(G\) that doesn't pass it? Answers use the Universal Frattini Cover (or minimal projective cover) of \(G\) from [\textit{M. Fried} and \textit{M. Jarden}, Field arithmetic (Ergebnisse 11) (1986; Zbl 0625.12001); Chapter 20]. Of course, the real algebraic numbers are close to the algebraic closure. The field of totally real numbers \(\mathbb{Q}^{tr}\) -- algebraic numbers all of whose conjugates are real -- gives closer comparison with regular realization of groups over \(\mathbb{Q}\). Theorem 5.7 uses [\textit{M. Fried} and \textit{H. Völklein}, Ann. Math., II. Ser. 135, 1-13 (1992; Zbl 0765.12002)] and [\textit{F. Pop}, Fields of totally \(\Sigma\)-adic numbers (Preprint 1991)] to show each finite group is the Galois group of a regular extension of \(\mathbb{Q}^{\text{tr}}(x)\). Theorem 5.7 follows trivially if we know \(G(\overline{\mathbb{Q}}/\mathbb{Q}^{\text{tr}})\) -- generated by the conjugate of complex conjugation -- is freely (in a profinite sense) generated by involutions. \textit{M. Fried}, \textit{D. Haran} and \textit{H. Völklein} [C. R. Acad. Sci., Paris, Sér. I 317, No. 11, 995-999 (1993)] have shown this. Pop has recently announced \(p\)-adic analogs of this based on techniques of Harbater. The second part of the paper, descends fields of definition from \(\mathbb{R}\) to \(\mathbb{Q}\). Hurwitz family techniques completely reduce the regular version of the inverse Galois problem to finding \(\mathbb{Q}\)-rational point on varieties [\textit{M. Fried} and \textit{H. Völklein}, Math. Ann. 290, 771-800 (1991; Zbl 0763.12004)]. One result considers realizing the symmetric group \(S_ m\) as the group of a Galois regular extension of \(\mathbb{Q}(x)\), satisfying two further conditions. These are that the extension has no more than four branch points, and it has some totally real residue class field specializations. Theorem 4.11 shows such extensions exist for \(m=3-10\). This topic demonstrates the diophantine difficulty of realizing groups by covers having quotients of \(G(\mathbb{Q}^{\text{tr}}/\mathbb{Q})\) as specializations. Finally, the paper considers the regular realization problem on the dihedral groups \(\{D_ p\), \(p\) a prime\}. Suppose \(p>7\). Theorem 5.1 shows if \(D_ p\) is the group of a Galois regular extension of \(\mathbb{Q}(x)\), the extension has at least 6 branch points. This interprets rational points on modular curves as coming from realization of certain dihedral group covers. It then applies Mazur's Theorem uniformly bounding \(\mathbb{Q}\) points on the modular curves \(\{C_ 1(p)\), \(p\) a prime\}. \textit{B. Mazur} and \textit{S. Kamienny} [Rational torsion of prime order in elliptic curves over number fields (Preprint 6/92)] suggest no bound on the number of branch points allows realization of more than finitely many \(D_ p\). Indeed, the Galois problem is harder than universally bounding elliptic curve torsion points over all extensions of \(\mathbb{Q}\) of a given degree.
Galois group of regular extension of \(\mathbb{Q}\)
regular realization problem on dihedral groups
scientific article; zbMATH DE number 537375