Definition:Generated Field Extension
Definition
Let $E / F$ be a field extension.
Let $S \subset E$ be a subset of $E$.
Definition 1
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 $S$.
Definition 2
Let $F \sqbrk {\set {X_s} }$ be the polynomial ring in $S$ variables $X_s$.
Let $\operatorname {ev} : F \sqbrk {\set {X_s} } \to E$ be the evaluation homomorphism associated to the inclusion $S \hookrightarrow E$.
The field extension $F \sqbrk S$ generated by $S$ is the set of all elements of $E$ of the form $\map {\operatorname {ev} } f / \map {\operatorname {ev} } g$, where $\map {\operatorname {ev} } g \ne 0$.
$S$ is said to be a generator of $F \sqbrk S$.