Topics in computational algebraic number theory

Belabas, Karim

doi:10.5802/jtnb.433

Cited by 41 publications

(47 citation statements)

References 29 publications

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“…All numerical results of the sections 6 and 9 have been calculated with the aid of programs which we have developed for the number theoretic computer algebra system PARI/GP, version 2.3.4 (2008) [5,34]. Details of our algorithms are presented in the related paper [28].…”

Section: Final Remarksmentioning

confidence: 99%

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Mayer¹

2012

Int. J. Number Theory

115

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Abstract. For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group G = Gal(F 2 p (K)|K) of the second Hilbert p-class field F 2 p (K) of K are determined by the p-class numbers of the unramified cyclic extensions N i |K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field K = Q( √ D) and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields L i |Q of absolute degree p of the dihedral fields N i . As an application, the structure of the automorphism group G = Gal(F 2 3 (K)|K) of the second Hilbert 3-class field F 2 3 (K) is analysed for all quadratic fields K with discriminant −10 6 < D < 10 7 and 3-class group of type (3, 3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3, r), 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated. IntroductionFor an algebraic number field K and a prime p ≥ 2 we denote by Cl p (K) the p-class group of K, that is the Sylow p-subgroup Syl p Cl(K) of its ideal class group. In this paper we shall be concerned with number fields having an elementary abelian bicyclic p-class group of type (p, p). We define the p-class field tower of The aim of the present paper is to investigate the second p-class group G = Gal(F 2 p (K)|K), that is the Galois group of the two-stage tower,, of a base field K with p-class group of type (p, p). The second p-class group G is distinguished by the special property that its commutator subgroupis isomorphic to the p-class group of the first Hilbert p-class field of K, whence G is metabelian.The theory of second p-class groups was initiated by Scholz and Taussky [38,39], using Schreier's concept of group extensions [36,37] Eick, et al. [11,10,12].

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Section: Final Remarksmentioning

confidence: 99%

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Mayer¹

2012

Int. J. Number Theory

115

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“…For most computational challenges we rely on well known techniques as described in [2,8,24] (with a few modifications). However, we use new strategies for ideal arithmetic and computations with approximations.…”

Section: Hecke: Algebraic Number Theory In Juliamentioning

confidence: 99%

Nemo/Hecke

Fieker

Hart

Hofmann

et al. 2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a lowlevel C implementation. For specialised algorithms, we use Julia's efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb, Antic and Singular. We give examples of how to use Hecke and Nemo and discuss some algorithms that we have implemented to provide high performance basic arithmetic. ACM Reference format:

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“…The quadratic form T 2 induces a natural lattice structure for O K (see for instance [15]) which can be…”

Section: Weighted List Decoding Of Nf-codesmentioning

confidence: 99%

“…, ω d . In particular, computing the coefficients requires performing polynomially (in d and z i ) many multiplications in K. The reader is referred to Belabas [15], Cohen [36], and Lenstra [107] for further details on arithmetic in number fields. For x, y ∈ O K , computing the coefficients of the product xy with respect to ω 1 , .…”

Section: Weighted List Decoding Of Nf-codesmentioning

confidence: 99%

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On the number field sieve: polynomial selection and smooth elements in number fields

Coxon¹

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The number field sieve is the asymptotically fastest known algorithm for factoring large integers that are free of small prime factors. Two aspects of the algorithm are considered in this thesis: polynomial selection and smooth elements in number fields. The contributions to polynomial selection are twofold.First, existing methods of polynomial generation, namely those based on Montgomery's method, are extended and tools developed to aid in their analysis. Second, a new approach to polynomial generation is developed and realised. The development of the approach is driven by results obtained on the divisibility properties of univariate resultants.Examples from the literature point toward the utility of applying decoding algorithms for algebraic error-correcting codes to problems of finding elements in a ring with a smooth representation. In this thesis, the problem of finding algebraic integers in a number field with smooth norm is reformulated as a decoding problem for a family of error-correcting codes called NF-codes. An algorithm for solving the weighted list decoding problem for NF-codes is provided. The algorithm is then used to find algebraic integers with norm containing a large smooth factor. Bounds on the existence of such numbers are derived using algorithmic and combinatorial methods.ii

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Topics in computational algebraic number theory

Cited by 41 publications

References 29 publications

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Nemo/Hecke

On the number field sieve: polynomial selection and smooth elements in number fields

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