Realization conjecture for finite groups as Galois groups of coverings

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Let k(x)k(x) be the function field of the projective line, and let EE be a finite regular Galois extension of k(x)k(x) with Galois group GG. Equivalently, the inclusion k(x)↪Ek(x)\hookrightarrow E defines a ramified Galois covering C→P1C\to\mathbb{P}^{1} over kk with Galois group GG. Realization conjecture. Every finite group GG occurs as the Galois group of such a covering. This is a realization problem for finite groups as Galois groups of regular function-field extensions, and the supplied text does not state whether the claim is known or open.

References

Primary source

Alberto Besana and Cristina Martinez, “Enumerative geometry of the curves defined by y^d=f(x)”, arXiv:1111.6456 (2012).

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