Abstract
The complete collection of cubic fields with a given fundamental discriminant can be constructed from certain algebraic integers in the associated quadratic resolvent field. Berwick explained how each such quadratic integer determines the roots of a cubic polynomial with rational coefficients. He referred to these elements as (quadratic) generators since they are generators of ideals in the maximal order of the quadratic resolvent field whose cubes are principal. In an unpublished manuscript, Shanks described how to incorporate efficient arithmetic on reduced ideals in the maximal order of the quadratic resolvent field into Berwick’s construction to produce irreducible cubic polynomials with very small coefficients. A computer implementation of Shanks’ method due to Fung found generating polynomials of all 364 cubic fields with a 19-digit discriminant. This chapter presents the never before published Shanks-Fung algorithm and, for completeness, concludes with a brief summary of Belabas’ fast technique for tabulating all cubic fields of bounded discriminant.
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Notes
- 1.
Algorithm (A), Corollary (C), Example (E), Figure (F), Lemma (L), Proposition (P), Remark (R), Theorem (T).
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Hambleton, S.A., Williams, H.C. (2018). Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler). In: Cubic Fields with Geometry. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-01404-9_4
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