Proposition 9

— Let L be an extension of a field K and M a commutative L-algebra (for example, an extension of L). If M is separable over L and L is separable over K, then M is separable over K.

Let K' be an extension of K. Since L is a separable extension of K, the ring $K' \otimes_K L$ is reduced; since M is a separable L-algebra, Prop. 5 (V, p. 120) shows the ring $(K' \otimes_K L) \otimes_L M$ to be reduced. Now the ring $K' \otimes_K M$ is isomorphic to $(K' \otimes_K L) \otimes_L M$ (II, p. 278, Prop. 2), hence is reduced. This proves that M is separable over K.

If the extension M is separable over K, it is not necessarily separable over L (cf. however V, p. 124, Cor. 3). For example, if p is a prime number, the field $F_p(X)$ of rational fractions in one indeterminate X over $F_p$ is separable over $F_p$ (V, p. 121, Prop. 6) but it is a p-radical algebraic extension of $F_p(X^p)$; in particular $F_p(X)$ is not separable over $F_p(X^p)$.

Later (V, p. 137 Prop. 5) we shall study the separability of composite extensions.