Small generators of the ideal class group |
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Authors: | Karim Belabas Francisco Diaz y Diaz Eduardo Friedman |
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Institution: | Université Bordeaux I, IMB--UMR 5251, 351 cours de la Libération, F-33405 Talence cedex, France ; Université Bordeaux I, IMB--UMR 5251, 351 cours de la Libération, F-33405 Talence cedex, France ; Departamento de Matemática, Universidad de Chile, Casilla 653, Santiago, Chile |
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Abstract: | Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group of a number field can be generated by the prime ideals of having norm smaller than . This result is essential for the computation of the class group and units of by Buchmann's algorithm, currently the fastest known. However, once has been computed, one notices that this bound could have been replaced by a much smaller value, and so much work could have been saved.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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Keywords: | Ideal class group generalized Riemann hypothesis |
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