proof of characterization of perfect fields
Proposition 1
The following are equivalent^{}:

(a)
Every algebraic extension^{} of $K$ is separable^{}.

(b)
Either $\mathrm{char}K=0$ or $\mathrm{char}K=p$ and the Frobenius map^{} is surjective^{}.
Proof. Suppose (a) and not (b). Then we must have $\mathrm{char}K=p>0$, and there must be $a\in K$ with no $p$th root in $K$. Let $L$ be a splitting field^{} over $K$ for the polynomial^{} ${X}^{p}a$, and let $\alpha \in L$ be a root of this polynomial. Then ${(X\alpha )}^{p}={X}^{p}{\alpha}^{p}={X}^{p}a$, which has coefficients in $K$. This means that the minimum polynomial for $\alpha $ over $K$ must be a divisor of ${(X\alpha )}^{p}$ and so must have repeated roots. This is not possible since $L$ is separable over $K$.
Conversely, suppose (b) and not (a). Let $\alpha $ be an element which is algebraic over $K$ but not separable. Then its minimum polynomial $f$ must have a repeated root, and by replacing $\alpha $ by this root if necessary, we may assume that $\alpha $ is a repeated root of $f$. Now, ${f}^{\prime}$ has coefficients in $K$ and also has $\alpha $ as a root. Since it is of lower degree than $f$, this is not possible unless ${f}^{\prime}=0$, whence $\mathrm{char}K=p>0$ and $f$ has the form:
$$f={x}^{pn}+{a}_{n1}{x}^{p(n1)}+\mathrm{\dots}+{a}_{1}{x}^{p}+{a}_{0}.$$ 
with ${a}_{o}\ne 0$. By (b), we may choose elements ${b}_{i}\in K$, for $0\le i\le n1$ such that $b_{i}{}^{p}={a}_{i}$. Then we may write $f$ as:
$$f={({x}^{n}+{b}_{n1}{x}^{n1}+\mathrm{\dots}+{b}_{1}x+{b}_{0})}^{p}.$$ 
Since $f(\alpha )=0$ and since the Frobenius map $x\mapsto {x}^{p}$ is injective^{}, we see that
$${\alpha}^{n}+{b}_{n1}{\alpha}^{n1}+\mathrm{\dots}+{b}_{1}\alpha +{b}_{0}=0$$ 
But then $\alpha $ is a root of the polynomial
$${x}^{n}+{b}_{n1}{x}^{n1}+\mathrm{\dots}+{b}_{1}x+{b}_{0}$$ 
which has coefficients in $K$, is nonzero (since ${b}_{o}\ne 0$), and has lower degree than $f$. This contradicts the choice of $f$ as the minimum polynomial of $\alpha $.
Title  proof of characterization of perfect fields 

Canonical name  ProofOfCharacterizationOfPerfectFields 
Date of creation  20130322 14:47:44 
Last modified on  20130322 14:47:44 
Owner  mclase (549) 
Last modified by  mclase (549) 
Numerical id  6 
Author  mclase (549) 
Entry type  Proof 
Classification  msc 12F10 