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In Lang's Algebra, I had hard times with this question:

Let $K$ be field with characteristic $p$ (a prime). Let $L|K$ be a finite extension of $K$, and suppose $gcd([L:K],p)=1$. Show $L$ is separable over $K$.

In order to have $L$ separable over $K$, one needs to have $K(\alpha)|K$ separable for each $\alpha \in L$.

We can say $L=K(\alpha_1,...,\alpha_m)$. It seems like I need to use the tower relation of these extensions and get some information about their degrees that will be relevant to the gcd thing. But I couldn't do these. A hint is welcomed.

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The extension $K(\alpha)/K$ is separable when the minimal polynomial of $\alpha$ over $K$ has no repeated roots.

An irreducible polynomial $f\in K[x]$ has repeated roots if and only if $f(x)=g(x^p)$ for some $g\in K[x]$.

Do you see how to proceed from here?

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  • $\begingroup$ An irreducible polynomial has repeated roots if and only if ... $\endgroup$ – Bernard Mar 24 '18 at 20:38
  • $\begingroup$ Ah, indeed - thanks for catching that! $\endgroup$ – Zev Chonoles Mar 24 '18 at 20:40
  • $\begingroup$ So, if $f$ has degree $n$, $g$ has degree $p^n$ which is equal to degree of $K(\alpha)|K$, leading to a contradiction. $\endgroup$ – offret Mar 24 '18 at 23:30

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