# Definition:Generator of Field

(Redirected from Definition:Generated 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 $K \cup S$.