Definition:Generated Field Extension/Definition 2

Definition
Let $E / F$ be a field extension.

Let $S \subset E$ be a subset of $E$.

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 $F \sqbrk S$.

Also see

 * Equivalence of Definitions of Generated Field Extension