A problem on field extension

Solution 1:

A field extension of finite degree has only finitely many intermediate extensions if and only if there is a primitive element. So if we can find a finite extension that has no primitive element then the number of intermediate fields must be infinite.

Consider $K = \mathbb F_p(X,Y)$, the field of rational functions in two variables over the finite field with $p$ elements, and the extension $L = K(X^{1/p}, Y^{1/p})$. Then $L$ is a field extension of degree $p^2$ over $K$. However, there cannot be a primitive element since $\alpha^p \in K$ for every $\alpha \in L$ but a primitive element must have degree $p^2$ over $K$.

The fundamental theorem of Galois theory doesn't apply here since the extension is not separable.

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EDIT: By looking at the proof of the primitive element theorem, an infinite number of intermediate fields can explicitly given by $K(\alpha X^{1/p}+Y^{1/p})$ for $\alpha \in K$. To show that these fields are distinct assume $E = K(\alpha X^{1/p}+Y^{1/p}) = K(\beta X^{1/p}+Y^{1/p})$ with $\alpha,\beta \in K$ and $\alpha \neq \beta$. Then $E$ contains $(\alpha-\beta)X^{1/p}$, hence it contains $X^{1/p}$ and $Y^{1/p}$. But then $E = K(X^{1/p},Y^{1/p})$ which is a contradiction since $\alpha X^{1/p}+Y^{1/p}$ would be a primitive element of $K(X^{1/p},Y^{1/p})$ over $K$ which is impossible.