Polynomials with Galois groups ${\rm Aut}(M\sb {22}),\;M\sb {22},$ and ${\rm PSL}\sb 3({\bf F}\sb 4)\cdot 2\sb 2$ over ${\bf Q}$

In this paper the construction of infinite families of polynomials with Galois groups $\operatorname{Aut} ({M_{22}})$, ${M_{22}}$ and ${\text{PSL}_3}({\mathbb{F}_4})\;\cdot\;2$ over $\mathbb{Q}$ 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.