Equations in characteristic 3 with Mathieu groups as their Galois groups

Abstract

Galois groups of some explicit equations in characteristic 3 will be computed to be the Mathieu group of degree 12. We use the resolution of singularities of plane curves to get a lower bound for the Galois groups. Also we use factorizations of polynomials over the prime field to sharpen the lower bound. Finally we use the ‘linearization process’ to get an appropriate upper bound for the Galois groups.