How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with defining polynomial $x^3ax^2(a+3)x 1$. Does anyone know of other families of totally real cubic fields for which the fundamental units are known? I'm particularly interested in cubic subfields of cyclotomic extensions.

1$\begingroup$ It can be shown that up to $\operatorname{GL}_2(\mathbb{Q})$ equivalence that polynomials of the shape $ax^3 + bx^2 + (b3a)x a$ are the only cubic polynomials with $C_3$ Galois group. Does this not suffice to show that all cubic subfields of cyclotomic extensions arise this way, via DeloneFadeev correspondence? $\endgroup$– Stanley Yao XiaoMay 17, 2016 at 13:47

$\begingroup$ Even in the cyclic case, a root of the Shanks polynomial (with integer parameter) does not necessarily generate the ring of integers of the field, and thus a pair of conjugate such roots does not necessarily give you a pair of fundamental units for the whole ring of integers  only for the subring generated by the root (and even this is nontrivial). $\endgroup$– GNiklaschMay 17, 2016 at 14:53
1 Answer
Here are two references that may be useful.
Algebraic number fields with 2 independent units, 1931, by W. E. H. Berwick.
The determination of units in totally real cubic fields, 1959, by H. J. Godwin.