Irreducibility of $x^4+x^2+x+1$ over $\mathbb{F}_5[x]$

Question

Consider

$$ f(x)=x^4+x^2+x+1\in \mathbb{F}_5[x] $$

There is a straightforward but lengthy proof that $f$ is irreducible: show that $f$ has no linear factors, then assume that it splits as a product of two quadratic factors and then derive a contradiction using the coefficients of $f$.

Is there a faster method for such a polynomial and maybe something that works for a larger class of polynomials?

Answer 1

Based on this answer you can use Rabin's test for irreducibility. In your case, you "just" need to show:

1. $x^4+x^2+x+1 | x^{5^4}-x$
2. $(x^4+x^2+x+1,x^{5^2}-x)=1$

You can use extended GCD to see that $$(x^4+x^2+x+1,x^{5^4}-x) = x^4+x^2+x+1$$

and also

$$(x^4+x^2+x+1,x^{5^2}-x)=1$$

so both conditions are satisfied. As is suggested in referenced link, computer algebra systems such as Mathematica/Maple can be used effectively.