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\(\widetilde M_{12}\) as Galois group over \({\mathbb Q})\). (\(\widetilde M_{12}\) comme groupe de Galois sur \({\mathbb Q}\).) (French) Zbl 0598.12008
This note is an illustration of Serre’s trace formula concerning the Witt invariant of the quadratic form \(\text{Tr}(X^ 2)\) [cf. J.-P. Serre, Comment. Math. Helv. 59, 651–676 (1984; Zbl 0565.12014)]. According to B. H. Matzat and A. Zeh-Marschke [J. Number Theory 23, 195–202 (1986; Zbl 0598.12007)], the Mathieu group \(M_{12}\) occurs as Galois group over the rational function field \({\mathbb Q}(T)\), and over \({\mathbb Q}\) for specializations of \(T\) in the set \(S=\{t\in {\mathbb Z}\mid t\equiv 1 \bmod 66\}\). One can then ask whether the same is true for the only nontrivial extension \(\widetilde M_{12}\) of \(M_{12}\) with kernel \({\mathbb Z}/2{\mathbb Z}\). The authors first show that the answer is negative over \({\mathbb Q}(T)\). Nevertheless, they prove, via Serre’s formula, that there exist specializations of \(T\) in \(S\) which conduce to realizations of \(\widetilde M_{12}\), and more generally to realizations of every central extension of \(M_{12}\), as Galois group over \({\mathbb Q}\).

11R32 Galois theory
12F12 Inverse Galois theory
11R34 Galois cohomology
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
20F29 Representations of groups as automorphism groups of algebraic systems
20D06 Simple groups: alternating groups and groups of Lie type
11S15 Ramification and extension theory