One out of the almost endless supply of identities discovered by Ramanujan is the following: $$ \sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$ which has the following interpretation in algebraic number theory: the fundamental unit $\sqrt[3]{2}-1$ of the pure cubic number field $K = {\mathbb Q}(\sqrt[3]{2})$ becomes a cube in the extension $L = K(\sqrt[3]{3})$.

Are there more examples of this kind in Ramanujan's work?