On the infrastructure of the principal ideal class of an algebraic number field of unit rank one

Abstract

Let R be the regulator and let D be the absolute value of the discriminant of an order O \mathcal {O} of an algebraic number field of unit rank 1. It is shown how the infrastructure idea of Shanks can be used to decrease the number of binary operations needed to compute R from the best known O ( R D ε ) O(R{D^\varepsilon }) for most continued fraction methods to O ( R 1 / 2 D ε ) O({R^{1/2}}{D^\varepsilon }) . These ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of O \mathcal {O} is principal.