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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)].

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