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Principal factors and lattice minima in cubic fields. (English) Zbl 1505.11140

Authors’ summary: “Let \(k = \mathbb{Q} (\sqrt[3]{d}, \zeta_3)\), where \(d >1\) is a cube-free positive integer, \(k_0 = \mathbb{Q}(\zeta_3)\) be the cyclotomic field containing a primitive cube root of unity \(\zeta_3\), and \(G=\operatorname{Gal}(k / k_0)\). The possible prime factorizations of \(d\) in our main result in our previous work [“Structure of relative genus fields of cubic Kummer extensions”, Preprint, arXiv:1808.04678, Theorem 1.1] give rise to new phenomena concerning the chain \(\Theta = (\theta_i)_{i\in\mathbb{Z}}\) of lattice minima in the underlying pure cubic subfield \(L = \mathbb{Q} (\sqrt[3]{d})\) of \(k\). The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals \((\nu) \in P_k^G/ P_{k_0}\) among the lattice minima \(\Theta = (\theta_i)_{i\in\mathbb{Z}}\) of the underlying pure cubic field \(L = \mathbb{Q} (\sqrt[3]{d})\), and to explain the exceptional behavior of the chain \(\Theta\) for certain radicands \(d\) with impact on determining the principal factorization type of \(L\) and \(k\) by means of Voronoi’s algorithm.”
Reviewer’s remarks: The decompositions of ideals, and describing its generators, are apparently more tricky and involved in fields \(\mathbb{Q}(\sqrt[3]{d},\zeta_3)\), where \(d>1\) is a cube-free positive integer, than in other fields. The authors meticulously dive into the details, thereby (for instance) correcting results by H. C. Williams [Math. Comput. 38, 261–274 (1982; Zbl 0491.12006)].
The paper is, however, rich of very much established details, worth to be studied.

MSC:

11R11 Quadratic extensions
11R16 Cubic and quartic extensions
11R20 Other abelian and metabelian extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R37 Class field theory

Citations:

Zbl 0491.12006
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Full Text: DOI arXiv

Online Encyclopedia of Integer Sequences:

Radicands of pure cubic number fields of type BETA and subtype M0.

References:

[1] S. Aouissi, D. C. Mayer, M. C. Ismaili, M. Talbi, and A. Azizi. 3-rank of ambiguous class groups of cubic Kummer extensions. Period. Math. Hungar. 81 (2019), 250-274. · Zbl 1474.11186
[2] S. Aouissi, A. Azizi, M. C. Ismaili, D. C. Mayer, and M. Talbi. Structure of relative genus fields of cubic Kummer extensions. Preprint, arXiv:1808.04678v2, 2019. · Zbl 1474.11186
[3] P. Barrucand and H. Cohn. A rational genus, class number divisibility, and unit theory for pure cubic fields. J.Number Theory 2(1) (1970), 7-21. · Zbl 0192.40001
[4] P. Barrucand and H. Cohn. Remarks on principal factors in a relative cubic field. J. Number Theory 3(2) (1971), 226-239. · Zbl 0218.12002
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[7] F. Gerth III. On 3-class groups of cyclic cubic extensions of certain number fields, J. Number Theory 8(1) (1976), 84-98. · Zbl 0329.12006
[8] G. F. Voronoi. Ob odnom obobshchenii algorithma nepreryvnykh drobei (On a generalization of the algorithm of continued fractions). Doctoral Dissertation, Warsaw, 1896 (in Russian).
[9] H. C. Williams. Improving the speed of calculating the regulator of certain pure cubic fields. Math. Comp. 35(152) (1980), 1423-1434. · Zbl 0455.12005
[10] H. C. Williams. Some results concerning Voronoi’s continued fraction over ℚ(3√D). Math. Comp. 36(154) (1981), 631-652. · Zbl 0486.12002
[11] H. C. Williams. Determination of principal factors in ℚ(√D) and ℚ(3√D). Math. Comp. 38(157) (1982), 261-274. · Zbl 0491.12006
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