splitting field

Let $f\in F[x]$ be a polynomial over a field $F$. A splitting field for $f$ is a field extension $K$ of $F$ such that

1. $f$ splits (factors into a product of linear factors) in $K[x]$,

2. $K$ is the smallest field with this property (any sub-extension field of $K$ which satisfies the first property is equal to $K$).

Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extension of the ground field.