Irreducible Polynomial in $\mathbb{F}_p (T) [x]$

Question

Let p be prime, and $K = {\bf F}_p (T)$ an extension of ${\bf F}_p$. How can i prove that the polynomial $x^p - T$ is irreducible in $K[x]$

Answer 1

Hint Since $F_p(T) = \operatorname{Quot}(F_p[T])$ and $x^p -T$ is a monic polynomial you may use Gauss‘ lemma to reduce to showing that $x^p-T$ is irreducible in $F_p[T][x]$, where it follows from Eisensteins criterion, because $T$ is prime.