## 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 $L$.)