Proposition 2

— For a field K to be perfect it is necessary and sufficient that every algebraic extension of K should be separable.

Suppose first that K is perfect. Since a field is a reduced ring, it follows from Lemma 5 ($V$, p. 35) that every extension of finite degree of $K$ is an etale algebra over $K$; therefore every algebraic extension of $K$ is separable.

Suppose now that $K$ is an imperfect field of characteristic $p \neq 0$. Let $\Omega$ be an algebraic closure of $K$. Since $K$ is imperfect, there exists $b \in K$ not belonging to $K^p$; put $a = b^{1/p}$. Then the extension $K(a)$ of $K$ is p-radical of finite degree. By $V$, p. 26, Prop. 3, there exists a single K-homomorphism of $K(a)$ into $\Omega$, and since $[K(a):K] > 1$, the algebra $K(a)$ is not etale over $K$ ($V$, p. 32, Prop. 4). In other words, the extension $K(a)$ of finite degree of $K$ is not separable.