For a given squarefree radicand R > 1, we determine the fundamental unit eta_{1} of the maximal order O_{1} in the real quadratic number field K = Q(R^{1/2}) with discriminant d = R (if R is congruent to 1 modulo 4) respectively d = 4R (if R is congruent to 2 or 3 modulo 4).
If a conductor f > 1 is given additionally, then we calculate also the fundamental unit eta_{f} of the order O_{f} with conductor f.
For this purpose, we utilize the continued fraction expansion (for the biggest integer) [1] of a suitable quadratic irrationality, that is, the quadratic VORONOI algorithm. Thus we run through a full period on the VORONOI highway of the appropriate order, i. e., in the linear chain of its lattice minima.
Then the regulators are the natural logarithms of the fundamental units, r_{1} = log(eta_{1}) and r_{f} = log(eta_{f}), and the index of the unit groups i = (U(O_{1}):U(O_{f})), for which eta_{f} = eta_{1}^{i}, coincides with the regulator quotient r_{f}/r_{1}.
Concerning the DIFFQI algorithm for the construction of dihedral discriminants with the aid of the regulator quotient criterion see [2].
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