\begin{definition}[Definition:Generated Field Extension/Definition 1]
Let $E / F$ be a field extension.
Let $S \subset E$ be a subset of $E$.
The '''field extension $F \sqbrk S$ generated by $S$''' is the smallest subfield extension of $E$ containing $S$, that is, the intersection of all subfields of $E$ containing $S$ and $F$.
Thus $S$ is a '''generator''' of $F \sqbrk S$ {{iff}} $F \sqbrk S$ has no proper subfield extension containing $S$.
\end{definition}