# Example of infinitely many intermediate field extensions

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?