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This question is from Kaplansky Fields and Rings 1.9.2

2. Let $K$ have characteristic $p$ and $L = K(u,v)$ where $u^p$, $v^p\in K$ and $[L:K] = p^2$. Show that $L$ is not a simple extension of $K$ and exhibit an infinite number of intermediate fields.

I am confused since the field $K$ could be finite. if $K$ is finite, then there are finitely many intermediate fields. I searched the internet, all related questions said the field $K$ and $L$ are function fields.

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A finite field extension that is not simple

G_D
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Takanashi
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1 Answers1

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The assumptions of the problem imply $K$ is not finite. Indeed, $x\mapsto x^p$ is a bijection from any finite field of characteristic $p$ to itself, so if $K$ were finite then $u^p,v^p\in K$ would imply $u,v\in K$ so $[L:K]$ would be $1$, not $p^2$.

Eric Wofsey
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