Existence of $p^{th}$ root in an inseparable extension

Question

Let $K= \mathbb{F}_p(t)$ i.e. the field of rational functions over $\mathbb{F}_p$.

I'm trying to prove that if $x$ is inseparable over $K$, then $K(x)$ contains a $p^{th}$ root of $t$.

I tried to find a $p^{th}$ root of $t$ by looking at the min poly of $x$ over $K$ as I know it will be in $K[T^p]$ but that didn't go anywhere. I'm not sure how I would use that $\min_K x$ contains a repeated root either.

I know I need to prove that $T^p-t\in K[T]$ splits over $K(x)$ and I thought about trying to use Kummer Theory but that won't work because the character of the field is the exponent of the polynomial.

I'm quite stuck so any help would be really appreciated.

Answer 1

If $x$ is inseparable then its minimal polynomial is $g(Y^p)$ with $g(Y)\in \Bbb{F}_p(t)[Y]$.

$g(Y^p) = h(Y)^p$ with $h(Y)\in \Bbb{F}_p(t^{1/p})[Y]$.

$$[\Bbb{F}_p(x,t^{1/p}):\Bbb{F}_p(t)] = p [\Bbb{F}_p(x,t^{1/p}):\Bbb{F}_p(t^{1/p})]\le p \deg(h)= \deg(g)=[\Bbb{F}_p(x,t):\Bbb{F}_p(t)]$$ which implies that $$\Bbb{F}_p(x,t^{1/p})=\Bbb{F}_p(x,t)$$