Corollary 1

— Let $A, B$ be two commutative algebras over $K$, of finite non-zero degrees. Then for the algebra $C = A \otimes_K B$ to be etale it is necessary and sufficient for $A$ and $B$ to be etale.

We have $[C: K] = [A: K] \cdot [B: K]$ and the corresponding Formula (7) for the separable degrees. Further we have $[A: K], \leq [A: K]$ and corresponding formulae for $B$ and $C$. It follows that $[C: K] = [C: K]$, if and only if we have both $[A: K] = [A: K]$, and $[B: K] = [B: K]$; now it is enough to apply Prop. 4.