Let \(K/k\) be an algebraic field extension. A normal closure of \(K/k\) is a field extension \(N/K\) such that \(N/k\) is normal and no proper subfield of \(N\) containing \(K\) is normal over \(k\).
A normal closure exists: inside an algebraic closure \(\bar K\) of \(K\), take the subfield generated over \(k\) by all roots in \(\bar K\) of the minimal polynomials over \(k\) of the elements of \(K\). It is unique up to an isomorphism that restricts to the identity on \(K\).