Topics in computational algebraic number theory

DOI10.5802/jtnb.433zbMath1078.11071OpenAlexW2091397771MaRDI QIDQ558179

Karim Belabas

Publication date: 30 June 2005

Published in: Journal de Théorie des Nombres de Bordeaux (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=JTNB_2004__16_1_19_0

zbMATH Keywords

surveyray class groupclass fielddiscrete logarithmscomputational methods for number fieldsPARI/GP

Mathematics Subject Classification ID

Research exposition (monographs, survey articles) pertaining to number theory (11-02) Algebraic number theory computations (11Y40)

Related Items (20)

Computation of lattice isomorphisms and the integral matrix similarity problem ⋮ Remarks on Euclidean minima ⋮ Quadratic p-ring spaces for counting dihedral fields ⋮ The anisotropic part of a quadratic form over a number field ⋮ THE SECOND p-CLASS GROUP OF A NUMBER FIELD ⋮ Algebraic number fields and the LLL algorithm ⋮ Multiparty non-interactive key exchange and more from isogenies on elliptic curves ⋮ Fast norm computation in smooth-degree abelian number fields ⋮ On the hardness of the NTRU problem ⋮ Reducing number field defining polynomials: an application to class group computations ⋮ On the computation of the HNF of a module over the ring of integers of a number field ⋮ List decoding of number field codes ⋮ Short Generators Without Quantum Computers: The Case of Multiquadratics ⋮ Short Bases of Lattices over Number Fields ⋮ Selected Applications of LLL in Number Theory ⋮ Short principal ideal problem in multicubic fields ⋮ Dependency of units in number fields ⋮ Computing $p$-adic $L$-functions of totally real number fields ⋮ Principalization algorithm via class group structure ⋮ Computing in quotients of rings of integers

Uses Software

• KANT/KASH
• PARI/GP
• ISOLATE

Cites Work

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• Generators and relations for \(K_{2}O_{F}\)
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• On computing subfields. A detailed description of the algorithm
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• KANT V4
• Subexponential algorithms for class group and unit computations
• A fast algorithm for polynomial factorization over \(\mathbb Q_p\)
• Efficient isolation of polynomial's real roots.
• Approximating rings of integers in number fields
• Optimizations of the subresultant algorithm
• A relative van Hoeij algorithm over number fields
• Computing class fields via the Artin map
• An Implementation of the Number Field Sieve
• Factor Refinement
• Improved Methods for Calculating Vectors of Short Length in a Lattice, Including a Complexity Analysis
• A Rigorous Subexponential Algorithm For Computation of Class Groups
• A polynomial reduction algorithm
• On factor refinement in number fields
• An efficient algorithm for the computation of Galois automorphisms
• Factoring into coprimes in essentially linear time
• Advanced Topics in Computional Number Theory
• Computing the Hilbert class field of real quadratic fields

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