Show $x^4+x^2+x+1$ is irreducible in $\mathbb{F}_5[x]$

Question

Show $x^4+x^2+x+1$ is irreducible in $\mathbb{F}_5[x]$.

I need to make use of Eisenstein's criterion, but I'm not sure how.

Answer 1

Hint: Let $x^4+x^2+x+1=(x^2+ax+c)(x^2+bx+d)$ and you search for $a,b,c,d\in F_5[x]$.

Answer 2

If $x^4+x^2+x+1$ were reducible over $\mathbb{F}_5$, it would have an irreducible factor with degree $\leq 2$. Since the polynomial $x^{25}-x$ is the product of all the irreducible monic polynomials over $\mathbb{F}_5$ with degree $1$ or $2$, it is enough to prove that over $\mathbb{F}_5$ we have $$\gcd(x^{25}-x,\,x^4+x^2+x+1)=1$$ to deduce that $x^4+x^2+x+1$ is an irreducible polynomial over $\mathbb{F}_5$. That is a completely algorithmic task.