It is well known that $\mathbb{F}_{p}(X,Y)/\mathbb{F}_{p}(X^{p},Y^{p})$ is not generated as a field extension by one primitive element, hence it has to have infinitely many intermediate fields of degree $p$ (assume $p$ a prime number) over $\mathbb{F}_{p}(X^{p},Y^{p})$. I think that such fields are of the form $\mathbb{F}_{p}(X^{p},Y^{p},f(X,Y))$ for some $f$ polynomial in two variables over $\mathbb{F}_{p}$. How do we show that such fields are infinitely many or how to concretely exhibit otherwise infinitely many intermediate subfields?


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