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1. Introduction
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 non-linear 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), Ex. I.10.2) that
three point ramified Galois extensions almost never have totally real specializations, for
example.
In this paper we give 2-, 3- and 4-parameter polynomials for certain (mostly nonsolvable) 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.
Acknowledgement: I would like to thank Peter M¨uller for very useful conversations on
the topic of this paper.

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