Proposition 12

— Let $E$ be an extension of $K$ and $E_s$, the set of elements of $E$ which are algebraic and separable over $K$. Then $E_s$, is the largest subextension of $E$ which is algebraic and separable over $K$.

By Prop. 6, a) ($V$, p. 39) every subextension of $E$ which is algebraic and separable over $K$ is contained in $E_s$. By Prop. 6, b) (loc. cit.) the extension $K(E_s)$ of $K$ is algebraic and separable, whence $K(E_s) \subset E$, and so $K(E_s) = E_s$.

With the notation of the preceding proposition, E, is called the relative separable (algebraic) closure of K in E. When K is perfect, E, is the relative algebraic closure of K in E (V, p. 36, Prop. 2).