Excuse my ignorance but how do you find the Galois group of an algebraic equation with integer coefficients modulo a prime number $ x^5+x+1\equiv 0\ \text{mod 5} $, say. Is there a Galois resolvent for this?
And if there is, how do you factorize an equation modulo prime, in case it hasn't got any of the p roots modulo prime, in quadratic and cubic degree polynomials or whatnot degree other than 1 say?
Many thanks.
In general, the splitting field of $f(X) \in \mathbb F_p[X]$ is $\mathbb F_{p^n}$, where $n$ is the smallest such that $f$ divides $X^{p^n}-X$ in $\mathbb F_p[X]$. Moreover, $\mathbb F_{p^n}$ is a cyclic extension of $\mathbb F_{p}$.