Subfield $K(\alpha)$ generated by an element of an extension field

Let \(F\) be a field, \(K\subseteq F\) a subfield, and \(\alpha\in F\). The subfield generated by \(K\) and \(\alpha\), written \(K(\alpha)\), is the smallest subfield of \(F\) containing \(K\) and \(\alpha\), that is, the intersection of all subfields of \(F\) that contain \(K\cup\theset{\alpha}\).