Let $K$ be a field of characteristic $p$ and $L=K(X,Y)$ where $X$ and $Y$ are variables (i.e. $L$ is the field of fractions of the polynomial ring $K[X,Y]$.
Let $\alpha,\beta\in\overline L$ such that $\alpha^p=X$ and $\beta^p=Y$. Let's consider the extension $L(\alpha,\beta)/L$. Clearly, $L(\alpha)/L$ and $L(\beta)/L$ are extensions of degree $p$. But what about the degree of $L(\alpha,\beta)/L\text{ ?}$ It seems natural that $T^p-X$ is still irreducible over $L(\beta)$, because we essentially don't add anything about $X$ when we go up to $L(\beta)$.
However, I haven't found an elegant/rigorous way to show that. What I actually did some time ago was writing $T^p-X$ as the product of two polynomials in $L(\beta)$ and observing that it led to something "strange", but it was quite messy and not totally rigorous.
Is there a nice way to prove that ?