Let \(F\) be the splitting field of \(f \in K[x]\) over \(K\). Prove that if \(g \in K[x]\) is irreducible and has a root in \(F\), then \(g\) splits into linear factors over \(F\).
Solution
A splitting field is a normal algebraic extension. Thus \(F/K\) is normal.
Let \(g\in K[x]\) be irreducible and suppose \(g\) has a root \(\alpha\in F\). By the defining irreducible-polynomial criterion for normality, every irreducible polynomial over \(K\) having one root in \(F\) splits completely over \(F\).
Applying this criterion to \(g\) gives \[ \boxed{g\text{ splits into linear factors in }F[x].} \]