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On the computation of quadratic 2-class groups. (English) Zbl 0870.11080
J. Théor. Nombres Bordx. 8, No. 2, 283-313 (1996); erratum ibid. 9, No. 1, 249 (1997).
The authors present a detailed description of an algorithm evolved from work of Gauss, D. Shanks, and 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.
Reviewer: H.Cohn (Bowie)

MSC:
11Y40 Algebraic number theory computations
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11E16 General binary quadratic forms
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
Software:
Magma
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References:
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