Let \(F\) be a field, \(K\subseteq F\) a subfield, and \(\alpha\in F\). The subring generated by \(K\) and \(\alpha\), written \(K[\alpha]\), is the smallest subring of \(F\) containing \(K\) and \(\alpha\), that is, the intersection of all subrings of \(F\) that contain \(K\cup\theset{\alpha}\).
\(K[\alpha]=\theset{g(\alpha) \st g\in K[x]}\), the image of the evaluation homomorphism \(K[x]\to F\), \(g\mapsto g(\alpha)\): this image is a subring containing \(K\) and \(\alpha\), and every such subring contains each \(g(\alpha)\).
For $\alpha$ in an extension $F \supset K$, what is $K[\alpha]$?