We present a very fast algorithm to build up tables of cubic fields. Real cubic fields with discriminant up to 10 and complex cubic fields down to −10 have been computed. The classification of… (More)

Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as Berlekamp-Zassenhaus, but uses lattice basis reduction to improve drastically on… (More)

J. Tate has determined the group K2OF (called the tame kernel) for six quadratic imaginary number fields F = Q( √ d), where d = −3,−4,−7, −8,−11, −15. Modifying the method of Tate, H. Qin has done… (More)

We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete… (More)

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with… (More)

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… (More)

In the last decades, a method has gained ever-increasing influence which treats deterministic objects as if they were random objects and studies them with probability theoretic means. A major… (More)

Assuming the Generalized Riemann Hypothesis, Bach has shown that the ideal class group C K of a number field K can be generated by the prime ideals of K having norm smaller than 12 ( log… (More)

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 been… (More)