Let \(L/k\) be an algebraic field extension. The extension \(L/k\) is normal if every irreducible polynomial \(f\in k[x]\) that has a root in \(L\) splits into linear factors in \(L[x]\).
Equivalently, for every \(\alpha\in L\), all roots of the minimal polynomial of \(\alpha\) over \(k\) in an algebraic closure of \(L\), the conjugates of \(\alpha\) over \(k\), lie in \(L\). Thus an irreducible \(f\in k[x]\) either splits in \(L[x]\) or has no root in \(L\).