# Definition:Generated Field Extension

## Definition

Let $E/F$ be a field extension.

Let $S \subset E$ be a subset.

### Definition 1

The field extension $F[S]$ 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 $F[S]$ if and only if $E$ has no proper subfield extension containing $S$.

### Definition 2

Let $F[\{X_s\}]$ be the polynomial ring in $S$ variables $X_s$.

Let $\operatorname{ev} : F[\{X_s\}] \to E$ be the evaluation homomorphism associated to the inclusion $S \hookrightarrow E$.

The field extension $F[S]$ generated by $S$ is the set of all elements of $E$ of the form $\operatorname{ev}(f)/\operatorname{ev}(g)$, where $\operatorname{ev}(g) \neq 0$, and $S$ is said to be a generator of $F[S]$.