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Let $K$ be a imperfect field. Let $L/K$ be a finite field extension.

Is $L$ imperfect?

Suppose that $L/K$ is separable. Is $L$ imperfect?

Suppose that $L/K$ is Galois. Is $L$ imperfect?

I'm looking for examples, counterexamples, explanations, and proofs of course.

Note: The converse holds. If $K$ is perfect, then any finite field extension of $K$ is perfect.

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A field is perfect if the separable closure is algebraically closed. So if a field is not perfect the separable closure is not algebraically closed. This says that if $L/K$ is separable then $L$ is imperfect. I'll think a little more about the general case. –  JSchlather Jan 23 '13 at 22:25

1 Answer 1

up vote 5 down vote accepted

Yes, if $K$ is imperfect then any finite extension $L$ of $K$ is also imperfect.
Indeed if $char.K=p$ and $a\in K$ has no $p$-th root in $K$, then any perfect field $P$ containing $K$ must contain all $a^{p^{-n}} \: (n\geq 1)$ and, since $[K(a^{p^{-n}}):K]=p^n$, the field $P$ must be of infinite dimension over $K$ (since its dimension is $\geq p^n$ for all $n$).
The fact that $L$ is separable or Galois over $K$ will not help: I'm sorry to say that $L$ will still be imperfect (but aren't we all...)

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