Realization of the Mathieu groups \(M_{11}\) and \(M_{12}\) as Galois groups over \({\mathbb Q})\).
The authors prove that the Mathieu groups \(M_{11}\) and \(M_{12}\) occur as Galois groups over \({\mathbb Q}(t)\). Moreover they construct a polynomial over \({\mathbb Q}(t)\) with Galois group \(M_{12}\) and compute specializations \(t\mapsto \tau \in {\mathbb Q}\) which realize \(M_{12}\) as Galois group over \({\mathbb Q}\). \(M_{12}\) is known to occur as Galois group over \({\mathbb Q}(t)\) and \({\mathbb Q}\) by work of the first author [J. Algebra 96, 499--531 (1985; Zbl 0587.12004)]. Using the same methods, but another class structure for \(M_{12}\) the present results on \(M_{12}\) and \(M_{11}\) are obtained; here the fixed field of the subgroup \(M_{11}\) of \(M_{12}=\text{Gal}(N/{\mathbb Q}(t))\) happens to be a rational function field \({\mathbb Q}(x)\). Constructing a polynomial for \(M_{12}\) leads to a nonlinear system of 13 equations in 12 unknowns over a quadratic extension field of \({\mathbb Q}\) which is solved on a computer with Buchberger's algorithm. A polynomial and specializations for \(M_{11}\) are announced by the authors.
\(M_{12}\) as Galois group over \({\mathbb Q}\)
scientific article; zbMATH DE number 3963969
scientific article