I am currently stuck on the following problem, taken from Herstein's Topics in Algebra, 2nd ed.. It reads:
If $F$ is of characteristic $p \neq 0$ and if $K$ is a finite extension of $F$, prove that given $a \in K$ either $a^{p^n} \in F$ for some $n$ or we can find an integer $m$ such that $a^{p^m} \notin F$ and is separable over $F$
Herstein defines an element to be separable over $F$ if, and only if, it is the root of a polynomial $p(x) \in F[x]$ with no multiple roots.
I am trying to do this by showing that, if we are not in the first case, we must be in the second case. However, I have no idea where to use that $K$ is a finite extension, and I just can't see why the second case must hold…
A couple of things I think could help are two former exercises from the same chapter. One says that the set $\{a \in K \mid a^{p^n} \in F\}$ is a subfield of $K$ and the other says that the set of separable elements of $K$ over $F$ are a subfield – but note that none of them suppose $K$ to be a finite extension…
I also thought about the fact that the polynomial $x^{p^n} – x$ has no multiple roots, but I don't know how to convert that to this problem.
In short, I am very much stuck and any help is appreciated!
Thanks in advance!
Best Answer
Some ideas for you: Let $\;a\in K\;$ and let $\;p_1(x)\in F[x]\;$ be its minimal polynomial over $\;F\;$ . If
$\;a\in F\;$ we're done since then any power of $\;a\;$ is again in $\;F\;$ . So assume $\;a\notin F\;$ . If $\;a\;$ is
separable then we're done, so assume $\;a\;$ isn't separable $\;\implies p_1(x)=g_1(x^p)\;$ (corollay 1), and
of course $\;\deg p_1>\deg g_1\;$.
The above means that $\;a^p\;$ is a root of $\;g_1(x)\;$ , and let $\;p_2(x)\in F[x]\;$ be the minimal polynomial
of $\;a^p\;$ over $\;F\;$ . If $\;a^p\in F\;$ we're thru, so assume $\;a^p\notin F\;$ . If $\;a^p\;$ is separable we're done, so
again assume $\;a^p\;$ is not separable $\;\implies p_2(x)=g_2(x^p)\;$ , and $\;\deg p_1>\deg g_1\ge\deg p_2>\deg g_2\;$
is a root of $\;g_2(x)\;$ and $\;(a^p)^p=a^{p^2}\;$...and continue on.
Since $\;\deg p_1\;$ is finite, the above must stop at some finite step...finish now the argument.