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In this paper the construction of infinite families of polynomials with Galois groups      Aut ⁡  ( M 22 ) \operatorname {Aut} ({M_{22}})      ,      M 22 {M_{22}}       and      PSL 3 ( F 4 ) ⋅  2 {\text {PSL}_3}({\mathbb {F}_4})\;\cdot \;2       over      Q \mathbb {Q}       is achieved. The determination of these polynomials leads to a system of nonlinear algebraic equations in 22 unknowns. The solutions belonging to the Galois extensions with the desired Galois groups are computed with a  p -modular version of the Buchberger algorithm. The application of this method, which is described in some detail, turns out to be feasible even for relatively large systems of nonlinear equations.

Polynomials with Galois groups 𝐴𝑢𝑡(𝑀₂₂),𝑀₂₂, and 𝑃𝑆𝐿₃(𝐹₄)⋅2₂ over 𝑄