Interactive Determination of the Regulator, Period Length, and Fundamental Unit of a Real Quadratic Field

Java applet with Quadratic VORONOI HiWay


For a given squarefree radicand R > 1, we determine the fundamental unit eta1 of the maximal order O1 in the real quadratic number field K = Q(R1/2) 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 etaf of the order Of 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, r1 = log(eta1) and rf = log(etaf), and the index of the unit groups i = (U(O1):U(Of)), for which etaf = eta1i, coincides with the regulator quotient rf/r1.

Concerning the DIFFQI algorithm for the construction of dihedral discriminants with the aid of the regulator quotient criterion see [2].


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