Multi-parameter Polynomials with Given Galois Group

The non-Abelian finite simple groups and their automorphism groups play a crucial role in an inductive approach to the inverse problem of Galois theory. The rigidity method (see, for example, Malle and Matzat, 1999) has proved very efficient for deducing the existence of Galois extensions with such groups, as well as for the construction of polynomials generating such extensions. Nevertheless, the effective construction requires the solution of a nonlinear system of equations, a problem which is known to be very hard from a complexity point of view. Thus, in practice, the computation of polynomials is restricted to rather small degree, to the case of stem fields of genus zero and also to few (mostly three) ramification points. For several applications, for example for the solution of embedding problems, it is sometimes necessary to find Galois extensions of the rationals with given group and with complex conjugation lying in a prescribed conjugacy class. But it is well known (see, for example, Malle and Matzat, 1999, Example I.10.2) that three-point ramified Galois extensions almost never have totally real specializations, for example. In this paper we give two-, threeand four-parameter polynomials for certain (mostly non-solvable) groups which, from a certain point of view, correspond to Galois extensions ramified in at least four points, with the property that these admit (infinitely many) totally real, Galois group preserving specializations. For example we obtain a two-parameter polynomial for the sporadic simple Mathieu group M12 over Q. Suitable specializations then yield totally real number fields with groups M11 and M12.