Definition:Generator of Field

Definition
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 $K \left[{S}\right]$, is the smallest subring of $F$ containing $K \cup S$.

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

Also see

 * Definition:Generator of Field Extension
 * Definition:Generated Field Extension
 * Subfield Generated by Subfield and Set equals Generated Field Extension