How do I prove that $\mathbb{F}_p(X^p,Y^p)\subset\mathbb{F}_p(X,Y)$ is not a primitive extension?
And how can I give infinitely many fields $E$ such that $\mathbb{F}_p(X^p,Y^p)\subset E\subset\mathbb{F}_p(X,Y)$?
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your second question is practically answered (for $p=2$) in http://math.stackexchange.com/questions/526000/intermediate-fields-between-mathbbz-2-sqrtx-sqrty-and-mathbbz-2 – mercio Mar 18 '16 at 15:17
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@mercio But that is specifically for $p=2$, how would I adapt it for general $p$? – Mar 18 '16 at 15:34
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This happens to be written on wikipedia: prove that any element in $\mathbb{F}_p(x,y)$ has degree $p$ over $\mathbb{F}_p(x^p,y^p)$.
