Let $\mathbb{F}_q$ be a finite field with $q=p^r$ elements where $p$ - prime. Let $f(x)\in \mathbb{F}_q[x]$ be an irreducible polynomial of degree $n$. Am I right that $\mathbb{F}_q[x]/\langle f(x)\rangle$ is the splitting field for $f(x)$?
P.S. I've proved it but would like to know is my observation correct?
$K=\mathbb{F}_q[x]/(f)$ is a $\mathbb{F}_q$ vector space of dimension $n$, it has $q^n$ elements. $K-\{0\}$ is a cyclic group of order $q^n-1$ so $K$ is the splitting field of $X^{p^n}-X$ since $f$ divides $X^{p^n}-X$ all the roots of $f$ are in $K$.