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A fast algorithm to compute cubic fields
  • K. Belabas
  • Mathematics, Computer Science
  • Math. Comput.
  • 1 July 1997
We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 10 11 and complex cubic fields down to -10 11 have been computed.
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User’s Guide to PARI / GP
  • 137
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A relative van Hoeij algorithm over number fields
  • K. Belabas
  • Computer Science, Mathematics
  • J. Symb. Comput.
  • 1 May 2004
Abstract van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as the Berlekamp–Zassenhaus algorithm, but uses lattice basis reduction toExpand
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The logarithmic class group package in PARI/GP
This note presents our implementation in the PARI/GP system of the various arithmetic invariants attached to logarithmic classes and units of number fields. Our algorithms simplify and improve onExpand
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Error estimates for the Davenport-Heilbronn theorems
We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of 3-torsion elements in theExpand
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Generators and relations for K_2 O_F.
Tate's algorithm for computing K_2 O_F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order---theExpand
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On quadratic fields with large 3-rank
  • K. Belabas
  • Computer Science, Mathematics
  • Math. Comput.
  • 30 January 2004
Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem,Expand
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Crible et 3-rang des corps quadratiques
Resume. Considerons le cardinal h∗ 3 (∆) de l’ensemble des racines cubiques de l’unite dans le groupe des classes de Q( √ ∆), ou ∆ est un discriminant fondamental. Un resultat de Davenport etExpand
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Topics in computational algebraic number theory
We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discreteExpand
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On the Mean 3-Rank of Quadratic Fields
The Cohen–Lenstra–Martinet heuristics give precise predictions about the class groups of a ‘random’ number field. The 3-rank of quadratic fields is one of the few instances where these have beenExpand
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