Definition 2

— *Let $E$ be an extension of $K$. Then $E$ is said to be quasi-Galois or normal (over $K$), if it is algebraic and if every irreducible polynomial of $K[X]$ which has at least one root in $E$, splits into a product of polynomials of degree 1 (distinct or not) in $E[X]$.

If $E$ is an algebraic closure of $K$, it is a quasi-Galois extension of $K$; for the condition (AC) of Prop. 1 ($V$, p. 19) asserts that every non-constant polynomial in $E[X]$ is a product of polynomials of degree 1.*