Definition:Generated Field Extension/Definition 1
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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 $S$.