On the computation of quadratic 2-class groups
The authors present a detailed description of an algorithm evolved from work of Gauss, D. Shanks, and \textit{J. C. Lagarias} [J. Algorithms 1, 142-186 (1980; Zbl 0473.68030)]. The idea is that the 2-class group for a discriminant \(D\) can be computed in cases where the full class group is not feasible, e.g., in random polynomial time, as long as the factors of \(D\) are known. The illustrations involve a \(D\) as a product of 5 primes close to \(10^{100}\). The prime factors immediately give ambiguous binary forms. Forms are successively examined by residue character for identification as a square. Then such a form is represented as (say) \(k^2\) (using ternary techniques), and the desired ``square-root-form'' is deduced from \((k^2,l,m)= 2(k,l,km)\). This is the detailed and time consuming part.
scientific article; zbMATH DE number 1002061
scientific article