\begin{definition}[Definition:Generator of Field]
Let $F$ be a field.
Let $S \subseteq F$ be a subset and $K \le F$ a subfield.
The '''field generated by $S$''' is the smallest subfield of $F$ containing $S$.
The '''subring of $F$ generated by $K \cup S$''', written $K \sqbrk S$, is the smallest subring of $F$ containing $K \cup S$.
The '''subfield of $F$ generated by $K \cup S$''', written $\map K S$, is the smallest subfield of $F$ containing $K \cup S$.
\end{definition}