Normal Extension

An algebraic field extension $L/K$ is said to be normal if $L$ is the splitting field of a family of polynomials in $K[X]$ .

Equivalent Properties
The normality of $L/K$ is equivalent to either of the following properties. Let $K^a$ be an algebraic closure of $K$ containing $L$ .

1) Every embedding $\sigma$ of $L$ in $K^a$ that restricts to the identity on $K$ , satisfies $\sigma(L)=L$ . In other words, $\sigma$ , is an automorphism of $L$ over $K$ .
2) Every irreducible polynomial in $K[X]$ that has one root in $L$ , has all of its roots in $L$ , that is, it decomposes into linear factors in $L[X]$ . (One says that the polynomial splits in $L$ .)