Computation of Galois groups

\(\(
<< "discrminant.m"\)\)

This notebook contains tools for investigation of Galois groups of rationalpolynomials of arbitray degree.  The basic tool used here is factorization of the reduction of an integer polynomial modulo a prime.  Fast algorithms are available for factorization of polynomials with coefficients in finite fields.  Mathematica makes available the command Factor[f, Modulus -> p]   to perform such factorizations.

Example:

f  =  x^5  - 5  x^4  +  2  x  - 1

Discriminant[f]Sqrt[Discriminant[f]]Factor[f,  Modulus -> 2]Factor[f,  Modulus -> 31]FactorTypes[f]