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    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?
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    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.
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    (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 p^n is appearing. The fields aren't necessarily of finite characteristic...?
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    (Original post by Zhen Lin)
    I'm confused about why p^n is appearing. The fields aren't necessarily of finite characteristic...?
    Yes they are.
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    (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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