I'm having problems factoring the following polynomial in $\mathbb{F}_5\left [ x \right ]$, someone could guide me please :'( $x^5+2x^4+3x^3+x^2+2x+4 \in \mathbb{F}_5\left [ x \right ]$.
Here is my attempt at what I have been able to understand: The idea is to use the algorithm of Cantor & Zasenhauss.
Let $f(x) := x^5+2x^4+3x^3+x^2+2x+4 \in \mathbb{F}_5\left [ x \right ] \Rightarrow f'(x) = 3x^3+4x^2+2x+2 \in \mathbb{F}_5\left [ x \right ]$. Then, we check that $f(x)$ is free of squares, ie that $gcd(f(x),f'(x))=1$, which is not verified, since $gcd(f(x),f'(x))=x+2$, therefore, we do $g(x) := \dfrac{f(x)}{gcd(f(x),f'(x))}=x^4+3x^2+2 \Rightarrow g'(x)=4x^3+x$ and so we get that $gcd(g(x),g'(x))=1$, ie $g(x)$ is free of squares.
From here I begin to complicate: (For the next steps I am guiding myself from here http://planetmath.org/cantorzassenhaussplit]http://planetmath.org/cantorzassenhaussplit
I will use the same notations on the attached page:
$B_1 = A, \ B_{k+1} := \displaystyle\frac{A}{gcd(B_k, x^{5^k}-x)} $
So, we have:
$B_1 := x^4+3x^2+2 $ $B_2 := \displaystyle\frac{x^4+3x^2+2}{gcd(B_k, x^5-x)}=x^3+2x^2+2x+4$ $B_3 := \displaystyle\frac{x^4+3x^2+2}{gcd(B_k, x^{25}-x)}=x^2+1$
And then how do I proceed? Can someone help me please? First of all, Thanks. Best regards.
I would start by looking for factors of degree $1$: it is not hard to find $x+3$ and $x+2$ (the latter with multiplicity $2$). After dividing out those, you're left with $x^2+2$ which is easily seen to be irreducible.