Ramanujan and algebraic number theory

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

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

• Dear Matt: $2^{1/3} - 1$ is a fundamental unit, so probably that is what Franz meant to write. – BCnrd Oct 24 '10 at 16:53

$$(7 \sqrt{20} - 19)^{1/6} = \ \sqrt{\frac{5}{3}} - \sqrt{\frac{2}{3}},$$ $$\left( \frac{3 + 2 \sqrt{5}}{3 - 2 \sqrt{5}} \right)^{1/4}= \ \ \frac{\sqrt{5} + 1}{\sqrt{5} - 1},$$ $$\left(\sqrt{\frac{1}{5}} + \sqrt{\frac{4}{5}}\right)^{1/2} = \ \ (1 + \sqrt{2} + \sqrt{8})^{1/5} = \ \ \sqrt{\frac{16}{125}} + \sqrt{\frac{8}{125}} + \sqrt{\frac{2}{125}} - \sqrt{\frac{1}{125}},$$ and so on. Many of these were submitted by Ramanujan as problems to the Journal of the Indian Mathematical Society. See the following link: http://www.math.uiuc.edu/~berndt/jims.ps for more precise references. Quote: "although Ramanujan never used the term unit, and probably did not formally know what a unit was, he evidently realized their fundamental properties. He then recognized that taking certain powers of units often led to elegant identities."
• I would have gone with 'enery  en.wikipedia.org/wiki/I'm_Henery_the_Eighth,_I_Am – Will Jagy Oct 25 '10 at 1:05