Definition:Generator of Field

Definition
Let $\left({F, +, \circ}\right)$ be a field.

Let $S \subseteq F$ be a subset and $K \leq F$ 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 $K[S]$, is the smallest subring of $F$ containing $K \cup S$.

The subfield of $F$ generated by $K \cup S$, written $K(S)$, is the smallest subfield of $F$ containing $K \cup S$.