Normal extensions are closed under compositum and intersection.Suggest a slogan
Let \(k\) be a field, let \(\Omega\) be an algebraic closure of \(k\), and let \(E_1, E_2 \subseteq \Omega\) be extensions of \(k\). If \(E_1/k\) and \(E_2/k\) are normal, then the compositum \(E_1E_2/k\) and the intersection \((E_1 \cap E_2)/k\) are normal.