Theorem 4

— Let $A$ be a commutative algebra of finite degree over $K$. Then the following assertions are equivalent: a) The algebra $A$ is etale. b) For every extension $L$ of $K$, the ring $L \otimes_K A$ is reduced. c) There exists a perfect extension field $P$ of $K$ such that the ring $P \otimes_K A$ is reduced. d) There exist separable algebraic extensions $L_1, \ldots, L_s$ of $K$ such that $A$ is isomorphic to $L_1 \times \cdots \times L_s$. In particular, every etale algebra is reduced.

A) Let us first prove the equivalence of $a$, $b$ and $c$.

Suppose that $A$ is etale and let $L$ be an extension of $K$. Let $\Omega$ be an algebraically closed extension field of $L$ (V, p. 23, Th. 2). Then $L \otimes_K A$ is isomorphic to a subring of $\Omega \otimes_K A$ and the latter is isomorphic to a ring $\Omega^n$ by Prop. 2 (V, p. 30). Therefore the ring $L \otimes_K A$ is reduced.

We have thus shown that a) implies b), and c) is a particular case of b). Assume now that c) holds. For the K-algebra $A$ to be etale it is necessary and sufficient that the P-algebra $A_{(P)}$ should be etale (V, p. 32, Cor. 2). The algebra $A$ is therefore etale by the following lemma: