L’algorithmique de la théorie algébrique des nombres
Journées mathématiques X-UPS, Théorie algorithmique des nombres et équations diophantiennes (2005), pp. 89-162.
Publié le :
DOI : 10.5802/xups.2005-02
Belabas, Karim 1

1 Université Bordeaux I, 351 Cours de la Libération, 33405 Talence Cedex, France
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Belabas, Karim. L’algorithmique de la théorie algébrique des nombres. Journées mathématiques X-UPS, Théorie algorithmique des nombres et équations diophantiennes (2005), pp. 89-162. doi : 10.5802/xups.2005-02. http://archive.numdam.org/articles/10.5802/xups.2005-02/

[1] Adleman, L. M.; Lenstra, H. W. Jr. Finding irreducible polynomials over finite fields, Proceedings of the eighteenth annual ACM symposium on Theory of Computing (Berkeley, CA) (STOC ’86), Association for Computing Machinery, New York, NY, 1986, pp. 350-355 | DOI

[2] Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin Primes is in P, Ann. of Math. (2), Volume 160 (2004) no. 2, pp. 781-793 | DOI | MR | Zbl

[3] Aho, A. V.; Hopcroft, J. E.; Ullman, J. D. The design and analysis of computer algorithms, Addison-Wesley, 1975

[4] Bach, E. Explicit bounds for primality testing and related problems, Math. Comput., Volume 55 (1990) no. 191, pp. 355-380 | DOI | MR | Zbl

[5] Beauzamy, Bernard Products of polynomials and a priori estimates for coefficients in polynomial decompositions : a sharp result, J. Symbolic Comput., Volume 13 (1992) no. 5, pp. 463-472 | DOI | MR | Zbl

[6] Belabas, Karim Topics in computational algebraic number theory, J. Théor. Nombres Bordeaux, Volume 16 (2004), pp. 19-63 | DOI | Numdam | MR | Zbl

[7] Belabas, Karim; van Hoeij, Mark; Klüners, Jürgen; Steel, Allan Factoring polynomials over global fields, J. Théor. Nombres Bordeaux, Volume 21 (2009) no. 1, pp. 15-39 http://jtnb.cedram.org/item?id=JTNB_2009__21_1_15_0 | DOI | Numdam | MR | Zbl

[8] Berlekamp, E. R. Factoring polynomials over large finite fields, Math. Comput., Volume 24 (1970), pp. 713-735 | DOI | MR | Zbl

[9] Buchmann, J.; Lenstra, H. W. Jr. Approximating rings of integers in number fields, J. Théor. Nombres Bordeaux, Volume 6 (1994) no. 2, pp. 221-260 | DOI | MR | Zbl

[10] Cohen, H.; Lenstra, H. W. Jr. Heuristics on class groups of number fields, Number theory (Noordwijkerhout 1983) (Lect. Notes in Math.), Volume 1068, Springer, Berlin, 1984, pp. 33-62 | MR | Zbl

[11] Cohen, H.; Martinet, J. Études heuristiques des groupes de classes des corps de nombres, J. reine angew. Math., Volume 404 (1990), pp. 39-76 | Zbl

[12] Cohen, Henri A course in computational algebraic number theory, Graduate Texts in Math., 138, Springer-Verlag, Berlin, 1993 | DOI

[13] Davenport, H.; Heilbronn, H. On the density of discriminants of cubic fields (II), Proc. Roy. Soc. London Ser. A, Volume 322 (1971), pp. 405-420 | MR | Zbl

[14] Demailly, J.-P. Analyse numérique et équations différentielles, Presses Universitaires de Grenoble, 1996

[15] Ellenberg, Jordan S.; Venkatesh, Akshay The number of extensions of a number field with fixed degree and bounded discriminant, Ann. of Math. (2), Volume 163 (2006) no. 2, pp. 723-741 | DOI | MR | Zbl

[16] Ford, David; Pauli, Sebastian; Roblot, Xavier-François A fast algorithm for polynomial factorization over p , J. Théor. Nombres Bordeaux, Volume 14 (2002) no. 1, pp. 151-169 | DOI | MR | Zbl

[17] von zur Gathen, Joachim; Gerhard, Jürgen Modern computer algebra, Cambridge University Press, Cambridge, 2013 | DOI

[18] Gourdon, Xavier Algorithmique du théorème fondamental de l’algèbre (1993) no. 1852 (Rapport de recherche)

[19] Hafner, J. L.; McCurley, K. S. A rigorous subexponential algorithm for computation of class groups, J. Amer. Math. Soc., Volume 2 (1989) no. 4, pp. 837-850 | DOI | MR | Zbl

[20] Hasse, H. Zahlentheorie, Akademie-Verlag GmbH, 1949

[21] Henrici, Peter Applied and computational complex analysis, Wiley-Interscience, New York, 1974 (Volume 1 : Power series—integration—conformal mapping—location of zeros)

[22] van Hoeij, Mark Factoring polynomials and the knapsack problem, J. Number Theory, Volume 95 (2002) no. 2, pp. 167-189 | DOI | MR | Zbl

[23] Lagarias, J. C.; Montgomery, H. L.; Odlyzko, A. M. A bound for the least prime ideal in the Chebotarev density theorem, Invent. Math., Volume 54 (1979) no. 3, pp. 271-296 | DOI | MR | Zbl

[24] Lagarias, J. C.; Odlyzko, A. M. Effective versions of the Chebotarev density theorem, Algebraic number fields : L-functions and Galois properties (Proc. Sympos., Durham, 1975), Academic Press, London, 1977, pp. 409-464 | Zbl

[25] Lang, Serge Algebraic number theory, Graduate Texts in Math., 110, Springer-Verlag, New York, 1994 | DOI

[26] The development of the number field sieve (Lenstra, A. K.; Lenstra, H. W. Jr., eds.), Lect. Notes in Math., 1554, Springer-Verlag, Berlin, 1993 | DOI | Zbl

[27] Lenstra, A. K.; Lenstra, H. W. Jr.; Lovász, L. Factoring polynomials with rational coefficients, Math. Ann., Volume 261 (1982) no. 4, pp. 515-534 | DOI | MR | Zbl

[28] Lenstra, H. W. Jr. Algorithms in algebraic number theory, Bull. Amer. Math. Soc. (N.S.), Volume 26 (1992) no. 2, pp. 211-244 | DOI | MR | Zbl

[29] Mignotte, M. An inequality about factors of polynomials, Math. Comput., Volume 28 (1974), pp. 1153-1157 | DOI | MR | Zbl

[30] Nguyen, Phong Q.; Stehlé, Damien Floating-point LLL revisited, Advances in cryptology—EUROCRYPT 2005 (Lect. Notes in Comput. Sci.), Volume 3494, Springer, Berlin, 2005, pp. 215-233 | DOI | MR | Zbl

[31] Oesterlé, J. Le problème de Gauss sur le nombre de classes, Enseign. Math., Volume 34 (1988), pp. 43-67 | MR | Zbl

[32] Papadimitriou, C. H. Computational complexity, Addison-Wesley, 1994

[33] Rouillier, F.; Zimmermann, P. Efficient isolation of polynomial’s real roots, J. Comput. Appl. Math., Volume 162 (2003) no. 1, pp. 33-50 | DOI | MR | Zbl

[34] Schoof, René Four primality testing algorithms, Algorithmic number theory : lattices, number fields, curves and cryptography (Math. Sci. Res. Inst. Publ.), Volume 44, Cambridge Univ. Press, Cambridge, 2008, pp. 101-126 | MR | Zbl

[35] Siksek, Samir The modular approach to Diophantine equations, Explicit methods in number theory (Panoramas & Synthèses), Volume 36, Société Mathématique de France, Paris, 2012, pp. 151-179 | MR | Zbl

[36] Stevenhagen, P.; Lenstra, H. W. Jr. Chebotarëv and his density theorem., Math. Intelligencer, Volume 18 (1996) no. 2, pp. 26-37 | DOI | Zbl

[37] Storjohann, A. Algorithms for matrix canonical forms, Ph. D. Thesis, ETH Zurich (2000)

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