Compute the degree of the splitting field

Question

I need to compute the degree of the splitting field of the polynomial $X^{4}+X^{3}+X^{2}+X+1$ over the field $\mathbb{F}_{3}$. Quite honestly I don't really know where to begin, I know the polynomial is irreducible in this field. So I thought we could consider some element $\alpha \in E$ where E is some field extensions of $\mathbb{F}_{3}$ and try to find a relation between $\alpha$ and the other roots, but I am not 100% sure. Also I think the degree is $4$ but I am not sure why. Any hints would be apprecaited. Please beare in mind that I am only a few weeks into my galois theory course so it might take a while for me to follow.

Answer 1

If $K$ is a finite field and $f \in K[X]$ is irreducible, then $K[X]/(f)$ is a splitting field of $f$. This follows from the fact that finite extensions of finite fields are always normal. If $\alpha$ is a root, the other roots are $\alpha^{p^n}$, $n \in \mathbb{N}$, where $p=\mathrm{char}(K)$. In particular, the degree is $\mathrm{deg}(f)$.

If $n$ is a natural number coprime to $p$, then it is a fact that the cyclotomic polynomial $\Phi_n$ is irreducible in $\mathbb{F}_p[x]$ if and only if $[p]$ generates $(\mathbb{Z}/n)^\times$.

Since $[3]$ generates $(\mathbb{Z}/5)^\times$, it follows that $\Phi_5=X^4+\dotsc+X+1$ is irreducible in $\mathbb{F}_3[X]$ and hence $\mathbb{F}_{3^4}$ is a splitting field. Explicitly, if $\alpha$ is a root of $\Phi_5$, then the other roots are $\alpha^2,\alpha^3,\alpha^4$.