A class \(\mathcal{S}\) of field extensions is distinguished if it satisfies the following two conditions.
- For every tower of fields \(k\subseteq K\subseteq L\), \(L/k \in \mathcal{S}\) if and only if \(L/K \in \mathcal{S}\) and \(K/k\in \mathcal{S}\):
- If \(K/k\in \mathcal{S}\) and \(L/k\) is an extension such that \(K\) and \(L\) are subfields of a common field, then \(LK/L \in \mathcal{S}\), where \(LK\) is the compositum. The extension \(LK/L\) is the lift of \(K/k\) to \(L\):
Let \(\mathcal{S}\) be a distinguished class of field extensions. If \(L/k\in\mathcal{S}\) and \(K/k\in \mathcal{S}\), with \(K\) and \(L\) subfields of a common field, then \(LK/k \in \mathcal{S}\):
By the lifting condition applied to \(K/k\in\mathcal{S}\) and \(L/k\), the extension \(LK/L\) lies in \(\mathcal{S}\). Applying the tower condition to \(k\subseteq L\subseteq LK\), with \(LK/L\in\mathcal{S}\) and \(L/k\in\mathcal{S}\), gives \(LK/k\in\mathcal{S}\).