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# Separable closure Watch

1. Let F/K be a finite extension. Show that there is a unique intermediate extension L/K such that L/K is separable and F/L is purely inseparable.

I've done some searching around and I think I understand the construction of L by taking the smallest subfield of F containing all the separable elements of F, but I can't figure out how to prove any properties of it. (I suspect this is because I failed to absorb very much of this section of the course.) Any hints, or references to good textbooks?
2. Presumably you can show that L/K is separable?

Consider the minimal polynomial f over L, of an element x in F-L. Prove that if it's separable it's linear, and that f(X) = g(X^p^n) with g separable.
3. (Original post by SimonM)
Presumably you can show that L/K is separable?

Consider the minimal polynomial f over L, of an element x in F-L. Prove that if it's separable it's linear, and that f(X) = g(X^p^n) with g separable.
I'm confused about why is appearing. The fields aren't necessarily of finite characteristic...?
4. (Original post by Zhen Lin)
I'm confused about why is appearing. The fields aren't necessarily of finite characteristic...?
Yes they are.
5. (Original post by SimonM)
Yes they are.
Ah, every algebraic extension of a characteristic zero field is separable. Right. I think I've got it now. Thanks!

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