Prove a necessary condition for the irreducibility of a polynomial over F_p

Question

Let $K=\mathbb{F}_p$, with $p$ prime, and $f(x)\in K[x]$ be a polynomial of degree $n>0$. Show that if $f(x)$ is irreducible then P has no roots in $\mathbb{F}_{p^d}$ with $d\le \frac{n}{2}$.

Answer 1

As $p$ is supposed to be irreducible, and $a$ is a root of $p$ in $\mathbb F_d$ we have

$$d=[\mathbb F_{p^d}: \mathbb F_p]=[\mathbb F_{p^d}:\mathbb F_p(a)][\mathbb F_p(a):\mathbb F_p]\ge n.$$

Hence $p$ can’t have a root in $\mathbb F_{p^d}$ if $d \le n/2$.