Proposition 2

— Let N be a Galois extension of K and E a subextension of N, of finite degree over K. There exists a subextension F of N containing E, Galois and of finite degree over K.

Since N is quasi-Galois over K, Cor. 1 of V, p. 56 proves the existence of a quasi-Galois subextension F of N containing E and of finite degree over K. Since N is separable over K, the same holds for F (V, p. 36, Prop. 1), hence F is Galois over K.

Prop. 2 implies the following result: let R be an algebraic closure of K and E₁, ..., Eₙ separable algebraic extensions of finite degree over K, contained in R. Then there exists a Galois extension N of K, of finite degree, contained in R and containing E₁, ..., Eₙ.