# Definition:Generated Field Extension/Definition 1

## Definition

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 $F \sqbrk S$ if and only if $F \sqbrk S$ has no proper subfield extension containing $S$.