Definition:Generated Field Extension

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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 $E$ 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 $F \sqbrk S$.


Also see