The Exercise 20.4 in Paul Garrett's Abstract Algebra states that
20.4 Let $x$ be an indeterminate over a field $k$ of characteristic $p$, a prime. Show that there are only finitely-many fields between $k(x)$ and $k\left(x^{1 / p}\right)$.
From my persepective, since $x^{1 / p}$ is a root of degree $p$ polynomial $f(y)=y^p-x$ over $k(x)$, we have $[k\left(x^{1 / p}\right):k(x)]\mid p$, which means $[k\left(x^{1 / p}\right):k(x)]=p$ as $p$ is a prime. Hence there does not exist an intermediate field $L$ satisfying $k\left(x^{1 / p}\right)\supsetneq L\supsetneq k(x)$ since otherwise, $[k\left(x^{1 / p}\right):L]>1,[L:k]>1$, contradicting $p$ being a prime.
I wonder whether my result that there are no intermediate field is correct, since the original exercise only requires to prove there are finitely-many fields in between.