Small generators of the ideal class group

Abstract

Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group CIK of a number field K can be generated by the prime ideals of K having norm smaller than 12(log |Discriminant(K)|)2 . This result is essential for the computation of the class group and units of K by Buchmann's algorithm, currently the fastest known. However, once CIK has been computed, one notices that this bound could have been replaced by a much smaller value, and so much work could have been saved. We introduce here a short algorithm which allows us to reduce Bach's bound substantially, usually by a factor 20 or so. The bound produced by the algorithm is asymptotically worse than Bach's, but favorable constants make it useful in practice. ©2007 American Mathematical Society.