## 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

Zbl 0473.68030

Magma
Full Text:

### References:

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