Proposition 4

— Let E be an extension of a field K; the set L of elements of E which are algebraic over K forms a subextension of E which is relatively algebraically closed in E.

For the field K(L) is algebraic over K (Cor. 1 of Th. 2), hence K(L) ⊂ L; it follows that K(L) = L and L is a field. On the other hand, if x ∈ E is algebraic over L, it is so also over K (Prop. 3) and hence belongs to L.

The extension L of K consisting of all the elements of E that are algebraic over K is called the relative algebraic closure of K in E; it is the largest algebraic extension of K contained in E.