Algebraic Extensions and Separability

I have been wrestling with the following problem for the past few hours, but I have made no progress whatsoever, namely:

Let $K/k$ be an algebraic extension with characteristic $p>0$ and let $\alpha \in K$. Show that $\alpha$ is separable over $k$ if and only if $k(\alpha^p)=k(\alpha)$. Conclude that every field extensions between finite fields is separable.

I wish I had progress to post, but, as of yet, I have made no progress beyond setting up the problem.

Could someone give a hint to push me in the right direction?

• Please see the following link: math.stackexchange.com/questions/126242/… – user38268 May 9 '12 at 14:25
• To those who wish to vote to close: Please do not close this question as the OP has an extra bit at the end that was different from mine. I just realised this but it is a bit too late to retract my own close vote. – user38268 May 9 '12 at 14:55

Ok let us see how to prove one direction, which I did not post in my question in that link. We would like to prove that if $\alpha$ is not separable over $k$, in other words if the minimal polynomial of $\alpha$ over $k$ is not separable then $k(\alpha^p) \subsetneqq k(\alpha)$. Now this is tantamount to proving that $\Big[k(a) : k(a^p)\Big] > 1$. Now

$$\Big[k(a) : k(a^p)\Big] = \frac{\Big[k(a):k\Big]}{\Big[k(a^p):k \Big]}.$$

The numerator on the right hand side is the degree of $f$ while the denominator is the degree of $g$, the minimal polynomial of $a^p$ over $F$. However since $f$ is not separable it is a polynomial in $x^p$. This is also saying that $f$ has $a^p$ as a root so that $g$ must divide $f$. It follows from here that $\deg g > \deg f$ so that

$$\Big[k[a]:k[a^p]\Big] > 1.$$

Can you prove the other direction?

Now for the last part of the problem in proving why every finite extension of a finite field is separable, we want that given any $\alpha$ in our extension $K$, the minimal polynomial $f$ of $\alpha$ over $K$ is separable. By your problem above, if we can show that

$$k(\alpha) = k(\alpha^p)$$

then we are done. Now one subset inclusion is already clear. For the other inclusion, we want to show that $k(\alpha) \subseteq k(\alpha^p)$. Now suppose that $[K:\Bbb{Z}/p\Bbb{Z}] = n$. This has to be finite in the first place because $K/k$ is a finite extension and $k/\Big(\Bbb{Z}/p\Bbb{Z}\Big)$ is a finite extension too. Then $K$ has $p^n$ elements so that its multiplicative group has $p^n-1$ elements. It follows by Lagrange's Theorem that

$$\alpha^{p^n - 1} = 1$$

so that for all $\alpha \in K$, we have that $\alpha^{p^n} = \alpha$. But then the left hand side can be written as $(\alpha^p)^{p^{n-1}}$ showing that $\alpha \in k(\alpha^p)$. Hence $k(\alpha) = k(\alpha^p)$ and the result follows.