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Finite groups as Galois groups of function fields with infinite field of constants. (English) Zbl 1194.12004

Given a field \(K\) and a finite group \(G\), the inverse Galois problem (for the pair \((K,G)\)) asks whether there exists a Galois extension of \(K\) whose Galois group is (isomorphic to) \(G\). In its full generality, this problem is still open, and in particular it is still open in the case when \(K = \mathbb{Q}\), the field of rational numbers. For this important case, Hilbert’s irreducibility criterion was devised to show that a positive answer for the pair \((\mathbb{Q},G)\) would follow from a positive answer for the pair \((\mathbb{Q}(x_1,\dots,x_n),G)\), where \(\mathbb{Q}(x_1,\dots,x_n)\) is the field of rational functions in many indeterminates over \(\mathbb{Q}\).
A weaker variant of the inverse Galois problem for the pair \((K,G)\) asks whether there exists an extension \(L\) of \(K\), not necessarily Galois, whose full automorphism group \(\operatorname{Aut}_K (L)\) over \(K\) is \(G\). A positive answer for the pair \((\mathbb{Q},G)\) follows immediately from the facts that every \(G\) is a subgroup of the symmetric group \(S_n\) for some \(n\) and that \(\mathbb{Q}\) has a Galois extension with Galois group \(S_n\), and using the fundamental theorem of Galois theory.
The paper under review is concerned with the afore-mentioned variant in the case when \(K\) is an algebraic function field over an infinite field \(k\) and with whether a positive answer for the pair \((K,G)\) would follow from a positive answer for the pair \((k(x),G)\), where \(k(x)\) is the field of rational functions in one indeterminate \(x\) over \(k\). It proves that if there exists an algebraic function field \(E\) over \(k\) such that (i) the extension \(E:k(x)\) is separable of degree greater than 1, (ii) there exists a place of degree 1 of \(k(x)\) that is ramified in \(E\), and (iii) \(\operatorname{Aut}_{k(x)}(E)\cong G\), then there are infinitely many nonisomorphic separable extensions \(L\) of \(K\) with \([L:K] = [E:k(x)]\) and \(\operatorname{Aut}_k(L) \cong G\).
Results that fall in this category were established by H. Stichtenoth in [Math. Z. 187, 221–225 (1984; Zbl 0527.12015)] and by M. Rzedowski-Calderón and G. Villa-Salvador in [Pac. J. Math. 150, 167–178 (1991; Zbl 0694.12011)]. Positive answers to the inverse Galois problem for \((K,G)\) when \(K\) is an algebraic function field and under certain restrictions were obtained by L. Greenberg in [Discontinuous Groups Riemann Surfaces, Proc. 1973 Conf. Univ. Maryland, 207–226 (1974; Zbl 0295.20053)], by D. J. Madden and R. C. Valentini in [J. Reine Angew. Math. 343, 162–168 (1983; Zbl 0502.12016)], by J. G. D’Mello and M. L. Madan in [Commun. Algebra 11, 1187–1236 (1983; Zbl 0531.12021)], and by M. Madan and M. Rosen in [Proc. Am. Math. Soc. 115, 923–929 (1992; Zbl 0791.14011)].

MSC:

12F12 Inverse Galois theory
11R58 Arithmetic theory of algebraic function fields
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20F29 Representations of groups as automorphism groups of algebraic systems
14H05 Algebraic functions and function fields in algebraic geometry
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References:

[1] DOI: 10.1080/00927878308822902 · Zbl 0531.12021 · doi:10.1080/00927878308822902
[2] Deuring, Lectures on the Theory of Algebraic Functions of One Variable (1973) · Zbl 0249.14008 · doi:10.1007/BFb0060944
[3] Álvarez-García, Int. J. Algebra 2 pp 65– (2008)
[4] DOI: 10.2307/2159335 · Zbl 0791.14011 · doi:10.2307/2159335
[5] DOI: 10.1007/BF01161706 · Zbl 0527.12015 · doi:10.1007/BF01161706
[6] Rzedowski-Calderón, Pacific J. Math. 150 pp 167– (1991) · Zbl 0694.12011 · doi:10.2140/pjm.1991.150.167
[7] Madden, J. Reine Angew. Math. 343 pp 162– (1983)
[8] Stichtenoth, Algebraic Function Fields and Codes (1993) · Zbl 1155.14022
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