Field \(\subseteq\) ring \(\subseteq\) field (D&F 13.2.16)

Let \(K/F\) be an algebraic extension and let \(R\) be a ring contained in \(K\) and containing \(F\). Show that \(R\) is a subfield of \(K\) containing \(F\).

By virtue of \(R\) being a subring of \(K\), many properties are automatically satisfied, such as the ring axioms, commutativity and 'integral-domain-ness'. The only remaining property to check is closure under inverses.

Let \(\alpha\) be a nonzero element of \(R\). Since \(\alpha\) is algebraic, its degree over \(F\) is finite. Each element of \(F(\alpha)\) is the form \(a_0 + a_1\alpha + \cdots + a_{n-1}\alpha^{n-1}\), where \(n\) is the degree of \(\alpha\) over \(F\). Thus \(F(\alpha) \subseteq R\) and so \(R\) contains \(\alpha^{-1}\) since \(F(\alpha)\) is a field.