Abstract
Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group C'K of a number eld K can be generated by the prime ideals of K having norm smaller than 12 logjDiscriminant(K)j 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 However, once C'K has been computed, one notices that this bound could have been replaced by a much smaller value, and so much work could have been saved.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.
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