# Polynomial factorisation over finite fields, part 2: Distinct degree factorisation

(Part of this.)

We now have our squarefree polynomial $A_i$, which is the product of all the factors of $A$ with exponent $i$. Our goal is to factor it into $A_i = A_{i,1}A_{i,2}\dots A_{i,\ell}$, where $A_{i,d}$ is the product of all the factors of $A_i$ of degree $d$. This is much simpler than the squarefree factorisation algorithm; it is essentially based on the fact that, over a field $\mathbf{F}_p$, the irreducible factors of $X^{p^n}-X$ are precisely the (monic) irreducible polynomials whose degree is a divisor of $n$.

# Polynomial factorisation over finite fields, part 1: Squarefree factorisation

(Continues this post.)

We wish to factor a polynomial $A$ with coefficients in the finite field $\mathbf{F}_p$ where $p$ is prime. As we stated in the previous post, the first step is to obtain a squarefree factorisation of $A$. Namely, we wish to obtain polynomials $A_1,A_2,\dots,A_k$, which are all squarefree and relatively prime, and such that $A = A_1A_2^2\dots A_k^k$. Ultimately, $A_i$ will be precisely the product of all the irreducible factors of $A$ with exponent $i$, which will ensure that all the above conditions are satisfied. We first need some preliminary results.