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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 $x^3-ax^2-(a+3)x -1$. 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.

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    $\begingroup$ It can be shown that up to $\operatorname{GL}_2(\mathbb{Q})$ equivalence that polynomials of the shape $ax^3 + bx^2 + (b-3a)x -a$ 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 Delone-Fadeev correspondence? $\endgroup$ – Stanley Yao Xiao May 17 '16 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). $\endgroup$ – GNiklasch May 17 '16 at 14:53
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Here are two references that may be useful.

  1. Algebraic number fields with 2 independent units, 1931, by W. E. H. Berwick.

  2. The determination of units in totally real cubic fields, 1959, by H. J. Godwin.

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