Corollary

— Let $A_1, \ldots, A_r$ be algebras over $K$ and $A = A_1 \times \cdots \times A_r$. For $A$ to be etale it is necessary and sufficient that $A_1, \ldots, A_r$ are etale.

Suppose that $A$ is etale; each of the algebras $A_1, \ldots, A_r$ is isomorphic to a quotient of $A$, and hence is etale by Prop. 3. Conversely, every extension of $K$ which diagonalizes $A_1, \ldots, A_r$ clearly diagonalizes $A$.