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Construction of number and function fields with given Galois group. (Konstruktion von Zahl- und Funktionenkörpern mit vorgegebener Galoisgruppe.) (German) Zbl 0555.12005
It is still an open question as to whether or not, given a finite group \(G\), there is a number field \(\Omega\) for which \(\mathrm{Gal}(\Omega/\mathbb{Q})\cong G\). If the field \(\mathbb{Q}\) is replaced by the rational function field \(\mathbb{C}(t)\), then Riemann’s Existence Theorem, combined with a classification due to Hurwitz, enables one essentially to give a complete solution to the analogous question. It is therefore natural to ask if those methods are applicable to the construction of finite extensions of \(\mathbb{Q}(t)\) or of \(\mathbb{Q}\) (or more generally of \(k(t)\) or \(k\), where \(k\) is a Hilbert subfield of \(\mathbb{C})\). Moreover, does every such Galois extension \(N/k(t)\), with Galois group \(G\), yield a Galois extension \(\Omega/k\), with group \(G\), by means of a suitable specialisation \(t\mapsto \tau\) for infinitely many \(\tau\in k\) ? The author considers these questions in this interesting paper.
He begins by investigating subfields \(k_0\) of \(\mathbb{C}\) that are fields of definition for algebraic function fields \(N/\mathbb{C}(t)\). For such fields, the product of the prime divisors of \(\mathbb{C}(t)\) that are ramified in \(N/\mathbb{C}(t)\) is defined over \(R_0 = k_0(t)\) and so it is natural to begin with the ramification structures of Galois extensions \(N/R\), \(R=k(t)\), \(k=\bar k_0\), the algebraic closure of \(k_0\) in \(\mathbb{C}\). The Galois extension \(N/R\) gives rise to an extension \(N_0/R_0\), which may not be Galois, but the Galois group \(G_0\) of the Galois closure contains a normal subgroup \(G_1\) isomorphic to \(G\) and by a proper choice of \(N_0\) one can ensure that \(G_0/G_1\cong \operatorname{Aut}(G)/\text{Inn}(G)\).
The author applies those results to the construction of Galois extensions of \(\mathbb{Q}(t)\) with given groups as Galois groups; for example, the alternating group \(A_n\), the groups \(\mathrm{PGL}(2,p)\) and \(\mathrm{PSL}(2,p)\), \(p\not\equiv \pm 1 \pmod{24}\), and the Mathieu groups \(M_{11}\) and \(M_{12}\).
The results proved here generalize theorems of K.-Y. Shih [Math. Ann. 207, 99–120 (1974; Zbl 0279.12102)]; M. Fried [Commun. Algebra 5, 17–82 (1977; Zbl 0478.12006)] and G. V. Belyǐ [Math. USSR, Izv. 14, 247–256 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267–276 (1979; Zbl 0409.12012)].
The fact that one can obtain Galois extensions of \(\mathbb{Q}(\sqrt{-11})\), \(\mathbb{Q}(\sqrt{-5})\) with Galois groups \(M_{11}\), \(M_{12}\) respectively, is established by the author in [Manuscr. Math. 27, 103–111 (1979; Zbl 0402.12003) and Arch. Math. 40, 245–254 (1983; Zbl 0494.12004)].

11R32 Galois theory
11R58 Arithmetic theory of algebraic function fields
12F12 Inverse Galois theory
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