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 0 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 0(RD£) for most continued fraction methods to 0(Rll2De).These ideas can also be applied to significantly decrease the number of operations needed to determine whether or not any fractional ideal of 0 is principal. 1. Introduction. In [16] Shanks introduced an idea which has since been modified and extended by Lenstra [13], Schoof [15] and Williams [17]. This idea can be used to decrease the number of binary operations needed to compute the regulator of a real quadratic order of discriminant D from 0(Dll2+e) to 0(D1/4+£) for every s > 0. In [21] and [17] Williams et al. showed that Shanks' idea could be extended to complex cubic fields.In this note we show that it can be further extended to any order 0 of an algebraic number field 7 of unit rank one, i.e., to orders of real quadratic, complex cubic, and totally complex quartic fields.We present an algorithm which computes the regulator R of 0 in 0(Rll2De) binary operations. Here, D is the absolute value of the discriminant of 0. We also describe a method for testing an arbitrary (fractional) ideal o of 0 for principality. This technique requires a number of binary operations that is 0(Rl/2De +p(m)), where p(m) is a polynomial in the input length m of a. The BabyStep Algorithm.Let 7 have degree n over the rationals Q, and suppose 7 has s real ¿¿-isomorphisms o\, <r2,..., os and t pairs of complex fi-isomorphisms cts+i,¿rs+1,... ,am,àm into C, m = s + t. Since 7 has unit rank 1, we have m = 2, and we have only two normalized Archimedian valuations on 7, | |i and | |2, where by |£|¿ we denote |cr¿(£)|e'. Here, e¿ = 1 when cr¿ is real and e¿ = 2 when oí is complex. As is usual in the unit theory, we introduce the logarithm mapping Log: 7X ^ Z £i->Logf = log|£|i.

Reduced Ideals in Pure Cubic Fields

“…A clear next step in generalization would be to address arbitrary cubic fields with negative discriminants; an integral basis of such fields is given, for example, in [3] and more briefly in [1]. Another possible generalization would be to arbitrary fields with rank one unit groups, including imaginary quartic fields, as in [4].…”

“…For each length between this minimum value and the upper bound, we examine each ideal. For each one, we obtain a list of pairs (y, z) satisfying conditions (1), ( 2), ( 3) and (4). For each such pair, we calculate P and Q and check our main condition from Theorem 2.9.…”

The reduced ideals of a special order in a pure cubic number field

“…Reduced ideals of a number field K form a finite and regularly distributed set in the infrastructure of K. They can be used to compute the regulator and the class number of a number field, see [5]- [10]. They can also be used to describe a method for testing an arbitrary (fractional) ideal for principality in algebraic number field of unit rank one, see [4].…”

Voronoi's algorithm in purely cubic congruence function fields of unit rank 1

“…Buchmann [1] generalized Voronoi's ideas to arbitrary number fields of unit rank 1 and 2. He extended his ideas to number fields of any rank [3,4] and subsequently incorporated the infrastructure concept in [6] and [5].…”