Construction of a regular extension of \(\mathbb{Q}(T)\) with Galois group \(M_{24}\). (Construction d’une extension régulière de \(\mathbb{Q}(T)\) de groupe de Galois \(M_{24}\).) (French) Zbl 0871.12006

It is proved that there exists a regular extension of \(\mathbb{K}(T)\) with Galois group the Mathieu group \(M_{23}\), where \(\mathbb{K}\) is a number field such that the equation \(x^2+ y^2+ z^2 =0\) has a nontrivial solution. The Mathieu group \(M_{23}\) is the only sporadic simple group which it is not yet known to occur as a Galois group over \(\mathbb{Q}\). B. H. Matzat [Publ., Math. Sci. Res. Inst. 16, 361-383 (1989; Zbl 0784.12005)] shows that the Mathieu group \(M_{24}\) appears as a Galois group of a regular extension of \(\mathbb{Q}(T)\), by proving the existence of a rational point in an appropriate Hurwitz space. The author performs such construction explicitly, replacing the usual tools of symbolic computation by numerical calculations. He determines the fixed field of degree 24 and genus 0 corresponding to the subgroup \(M_{23}\) and he shows that this fixed field is rational whenever the curve \(x^2+ y^2+ z^2 =0\) has \(\mathbb{K}\)-rational points.
Reviewer: N.Vila (Barcelona)


12F12 Inverse Galois theory
12Y05 Computational aspects of field theory and polynomials (MSC2010)


Zbl 0784.12005


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