<s>[INST] Ramanujan and algebraic number theory One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\rule{0pt}{2ex}\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
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?
 [/INST] $$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}}
- \sqrt[3]{\frac{2}{3}},$$
$$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}}
\right)^{1/4}= \  \  \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$
$$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2}
=  \  \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} =  \ \ 
\sqrt[5]{\frac{16}{125}} + \sqrt[5]{\frac{8}{125}} + \sqrt[5]{\frac{2}{125}} - \sqrt[5]{\frac{1}{125}},$$
and so on. Many of these were submitted by Ramanujan as problems to the
Journal of the Indian Mathematical Society. See the following link:
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."
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