Algebraic Number Fields

  • Olivier Bordellès
Part of the Universitext book series (UTX)


This chapter is an introduction to algebraic number fields, which arose from both a generalization of the arithmetic in ℤ and the necessity to solve certain Diophantine equations. After recalling basic concepts from algebra and providing some polynomial irreducibility tools, the ring of integers \(\mathcal {O}_{\mathbb {K}}\) of an algebraic number field \(\mathbb {K}\) is investigated. Next, the ideal numbers, as Kummer called them, are introduced to restore unique factorization. The last section shows how analytic tools can be used to solve hard problems of algebraic number theory. In particular, an account of Zimmert’s method for ideal classes is given.


  1. [Alz00]
    Alzer H (2000) Inequalities for the Gamma function. Proc Am Math Soc 128:141–147 MathSciNetzbMATHCrossRefGoogle Scholar
  2. [ASV79]
    Anderson N, Saff EB, Varga RS (1979) On the Eneström–Kakeya theorem and its sharpness. Linear Algebra Appl 28:5–16 MathSciNetzbMATHCrossRefGoogle Scholar
  3. [AW04a]
    Alaca S, Williams KS (2004) Introductory algebraic number theory. Cambridge University Press, Cambridge zbMATHGoogle Scholar
  4. [AW04b]
    Alaca S, Williams KS (2004) On Voronoï’s method for finding an integral basis of a cubic field. Util Math 65:163–166 MathSciNetzbMATHGoogle Scholar
  5. [Bak66]
    Baker A (1966) Linear forms in the logarithms of algebraic numbers. Mathematika 13:204–216 CrossRefGoogle Scholar
  6. [Bar78]
    Bartz KM (1978) On a theorem of Sokolovskii. Acta Arith 34:113–126 MathSciNetzbMATHGoogle Scholar
  7. [Bor02]
    Bordellès O (2002) Explicit upper bounds for the average order of d n(m) and application to class number. JIPAM J Inequal Pure Appl Math 3, Art. 38 MathSciNetGoogle Scholar
  8. [Bou70]
    Bourbaki N (1970) Algèbre I, chapitres 1 à 3. Hermann, Paris Google Scholar
  9. [CN63]
    Chadrasekharan K, Narasimhan R (1963) The approximate functional equation for a class of zeta-functions. Math Ann 152:30–64 MathSciNetCrossRefGoogle Scholar
  10. [Coh93]
    Cohen H (1993) A course in computational algebraic number theory. GTM, vol 138. Springer, Berlin zbMATHGoogle Scholar
  11. [Coh00]
    Cohen H (2000) Advanced topics in computational algebraic number theory. GTM, vol 193. Springer, Berlin CrossRefGoogle Scholar
  12. [Coh07]
    Cohen H (2007) Number theory Volume I: Tools and Diophantine equations. GTM, vol 239. Springer, Berlin Google Scholar
  13. [Cox89]
    Cox DA (1989) Primes of the form x 2+ny 2. Wiley, New York Google Scholar
  14. [CR00]
    Cohen H, Roblot X-R (2000) Computing the Hilbert class field of real quadratic fields. Math Comput 69:1229–1244 MathSciNetzbMATHGoogle Scholar
  15. [Cus84]
    Cusick TW (1984) Lower bounds for regulators. In: Number theory. Lect. notes math., vol 1068, pp 63–73. Proc J Arith, Noordwijkerhout/Neth 1983 Google Scholar
  16. [dlM01]
    de la Maza A-C (2001) Bounds for the smallest norm in an ideal class. Math Comput 71:1745–1758 CrossRefGoogle Scholar
  17. [EGP00]
    Elezović N, Giordano C, Pečarić J (2000) The best bounds in Gautschi’s inequality. Math Inequal Appl 3:239–252 MathSciNetzbMATHGoogle Scholar
  18. [EM99]
    Esmonde J, Murty MR (1999) Problems in algebraic number theory. GTM, vol 190. Springer, Berlin zbMATHCrossRefGoogle Scholar
  19. [Erd34]
    Erdős P (1934) A theorem of Sylvester and Schur. J Lond Math Soc 9:282–288 CrossRefGoogle Scholar
  20. [Erd53]
    Erdős P (1953) Arithmetic properties of polynomials. J Lond Math Soc 28:416–425 CrossRefGoogle Scholar
  21. [ESW07]
    Eloff D, Spearman BK, Williams KS (2007) A 4-sextic fields with a power basis. Missouri J Math Sci 19:188–194 zbMATHGoogle Scholar
  22. [FPS97]
    Flynn EV, Poonen B, Schaefer EF (1997) Cycles of quadratic polynomials and rational points on a genus-2 curve. Duke Math J 90:435–463 MathSciNetzbMATHCrossRefGoogle Scholar
  23. [Fri89]
    Friedman E (1989) Analytic formulas for the regulator of a number field. Invent Math 98:599–622 MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Fri07]
    Friedman E (2007) Regulators and total positivity. Publ Mat 51:119–130 zbMATHGoogle Scholar
  25. [FT91]
    Fröhlich A, Taylor MJ (1991) Algebraic number theory. Cambridge studies in advanced mathematics, vol 27. Cambridge University Press, Cambridge Google Scholar
  26. [Gar81]
    Garbanati D (1981) Class field theory summarized. Rocky Mt J Math 11:195–225 MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Gau86]
    Gauss CF (1986) Disquisitiones Arithmeticæ. Springer-Verlag, Berlin. Reprint of the Yale University Press, New Haven, 1966 zbMATHGoogle Scholar
  28. [GG04]
    Gras G, Gras M-N (2004) Algèbre Fondamentale, Arithmétique. Ellipses, Paris Google Scholar
  29. [Gol85]
    Goldfeld D (1985) Gauss’ class number problem for imaginary quadratic fields. Bull Am Math Soc 13:23–37 MathSciNetzbMATHCrossRefGoogle Scholar
  30. [Gra74]
    Gras M-N (1974) Nombre de classes, unités et bases d’entiers des extensions cubiques cycliques de ℚ. Mem SMF 37:101–106 MathSciNetzbMATHGoogle Scholar
  31. [Gre74]
    Greenberg M (1974) An elementary proof of the Kronecker–Weber theorem. Am Math Mon 81:601–607. Correction (1975), 81:803 zbMATHCrossRefGoogle Scholar
  32. [GT95]
    Gras M-N, Tanoé F (1995) Corps biquadratiques monogènes. Manuscr Math 86:63–79 zbMATHCrossRefGoogle Scholar
  33. [Győ76]
    Győry K (1976) Sur les polynômes à coefficients entiers et de discriminant donné, III. Publ Math (Debr) 23:141–165 Google Scholar
  34. [Hag00]
    Hagedorn TR (2000) General formulas for solving solvable sextic equations. J Algebra 233:704–757 MathSciNetzbMATHCrossRefGoogle Scholar
  35. [Has30]
    Hasse H (1930) Arithmetischen Theorie der kubischen Zahlkörper auf klassenkörper-theoretischer Grundlage. Math Z 31:565–582 MathSciNetzbMATHCrossRefGoogle Scholar
  36. [Hee52]
    Heegner K (1952) Diophantische Analysis und Modulfunktionen. Math Z 56:227–253 MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Her66]
    Herz CS (1966) Construction of class fields. In: Seminar on complex multiplication. Lecture notes in math., vol 21. Springer, Berlin, pp VII-1–VII-21 CrossRefGoogle Scholar
  38. [Hin08]
    Hindry M (2008) Arithmétique. Calvage & Mounet, Paris zbMATHGoogle Scholar
  39. [IK04]
    Iwaniec H, Kowalski E (2004) Analytic number theory. Colloquium publications, vol 53. Am. Math. Soc., Providence zbMATHGoogle Scholar
  40. [Jan96]
    Janusz G (1996) Algebraic number fields. Graduate studies in mathematics, vol 7, 2nd edn. Amer. Math. Soc., Providence zbMATHGoogle Scholar
  41. [Lan03]
    Landau E (1903) Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math Ann 56:645–670 MathSciNetzbMATHCrossRefGoogle Scholar
  42. [Lan27]
    Landau E (1927) Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. Teubner, Leipzig. 2nd edition: Chelsea, 1949 zbMATHGoogle Scholar
  43. [Lan93]
    Lang S (1993) Algebra, 3rd edn. Addison-Wesley, Reading zbMATHGoogle Scholar
  44. [Lan94]
    Lang S (1994) Algebraic number theory. GTM, vol 110, 2nd edn. Springer, Berlin zbMATHGoogle Scholar
  45. [Leb07]
    Lebacque F (2007) Generalized Mertens and Brauer–Siegel theorems. Acta Arith 130:333–350 MathSciNetzbMATHCrossRefGoogle Scholar
  46. [LN83]
    Llorente P, Nart E (1983) Effective determination of the decomposition of the rational primes in a cubic field. Proc Am Math Soc 87:579–582 MathSciNetzbMATHCrossRefGoogle Scholar
  47. [Lou98]
    Louboutin S (1998) Majorations explicites du résidu au point 1 des fonctions zêta de certains corps de nombres. J Math Soc Jpn 50:57–69 MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Lou00]
    Louboutin S (2000) Explicit bounds for residues of Dedekind zeta functions, values of L-functions at s=1, and relative class number. J Number Theory 85:263–282 MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Lou03]
    Louboutin S (2003) Explicit lower bounds for residues of Dedekind zeta functions at s=1 and relative class number of CM-fields. Trans Am Math Soc 355:3079–3098 MathSciNetzbMATHCrossRefGoogle Scholar
  50. [LSWY05]
    Lavallee MJ, Spearman BK, Williams KS, Yang Q (2005) Dihedral quintic fields with a power basis. Math J Okayama Univ 47:75–79 MathSciNetzbMATHGoogle Scholar
  51. [MO07]
    Murty MR, Order JV (2007) Counting integral ideals in a number fields. Expo Math 25:53–66 MathSciNetzbMATHCrossRefGoogle Scholar
  52. [Mol99]
    Mollin RA (1999) Algebraic number theory. Chapman and Hall/CRC, London zbMATHGoogle Scholar
  53. [Nar04]
    Narkiewicz W (2004) Elementary and analytic theory of algebraic numbers. SMM, 3rd edn. Springer, Berlin zbMATHGoogle Scholar
  54. [Neu10]
    Neukirch J (2010) Algebraic number theory. A series of comprehensive studies in mathematics, vol 322. Springer, Berlin Google Scholar
  55. [Odl76]
    Odlyzko AM (1976) Lower bounds for discriminants of number fields. Acta Arith 29:275–297 MathSciNetzbMATHGoogle Scholar
  56. [Poh77]
    Pohst M (1977) Regulatorabschätzungen für total reelle algebraische Zahlkörper. J Number Theory 9:459–492 MathSciNetzbMATHCrossRefGoogle Scholar
  57. [Poi77]
    Poitou G (1977) Minorations de discriminants (d’après AM Odlyzko). Séminaire Bourbaki, vol 1975/76, 28ème année. Springer, Berlin, pp 136–153 Google Scholar
  58. [Pra04]
    Prasolov VV (2004) Polynomials. ACM, vol 11. Springer, Berlin zbMATHGoogle Scholar
  59. [Ram01]
    Ramaré O (2001) Approximate formulæ for L(1,χ). Acta Arith 100:245–266 MathSciNetzbMATHCrossRefGoogle Scholar
  60. [Rem32]
    Remak R (1932) Über die Abchätzung des absoluten Betrages des Regulator eines algebraisches Zahlkörpers nach unten. J Reine Angew Math 167:360–378 Google Scholar
  61. [Rem52]
    Remak R (1952) Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers. Compos Math 10:245–285 MathSciNetzbMATHGoogle Scholar
  62. [Rib88]
    Ribenboim P (1988) Euler’s famous prime generating polynomial and the class number of imaginary quadratic fields. Enseign Math 34:23–42 MathSciNetzbMATHGoogle Scholar
  63. [Rib01]
    Ribenboim P (2001) Classical theory of algebraic numbers. Universitext. Springer, Berlin zbMATHGoogle Scholar
  64. [Ros94]
    Rose HE (1994) A course in number theory. Oxford Science Publications, London zbMATHGoogle Scholar
  65. [Sam71]
    Samuel P (1971) Théorie Algébrique des Nombres. Collection Méthodes. Hermann, Paris zbMATHGoogle Scholar
  66. [Sil84]
    Silverman J (1984) An inequality relating the regulator and the discriminant of a number field. J Number Theory 19:437–442 MathSciNetzbMATHCrossRefGoogle Scholar
  67. [Soi81]
    Soicher L (1981) The computation of Galois groups. PhD thesis, Concordia Univ., Montréal Google Scholar
  68. [Sok68]
    Sokolovskiǐ AV (1968) A theorem on the zeros of the Dedekind zeta function and the distance between neighbouring prime ideals (Russian). Acta Arith 13:321–334 MathSciNetGoogle Scholar
  69. [ST02]
    Stewart I, Tall D (2002) Algebraic number theory and Fermat’s last theorem, 3rd edn. AK Peters, Wellesley zbMATHGoogle Scholar
  70. [Sta67]
    Stark HM (1967) A complete determination of the complex quadratic fields of class-number one. Mich Math J 14:1–27 zbMATHCrossRefGoogle Scholar
  71. [Sta74]
    Stark HM (1974) Some effective cases of the Brauer–Siegel theorem. Invent Math 23:123–152 CrossRefGoogle Scholar
  72. [Sta75]
    Stark HM (1975) The analytic theory of algebraic numbers. Bull Am Math Soc 81:961–972 zbMATHCrossRefGoogle Scholar
  73. [Sta79]
    Stas W (1979) On the order of the Dedekind zeta-function near the line σ=1. Acta Arith 35:195–202 MathSciNetzbMATHGoogle Scholar
  74. [SW88]
    Schoof R, Washington LC (1988) Quintic polynomials and real cyclotomic fields with large class number. Math Comput 50:543–556 MathSciNetzbMATHGoogle Scholar
  75. [SWW06]
    Spearman BK, Watanabe A, Williams KS (2006) PSL(2,5) sextic fields with a power basis. Kodai Math J 29:5–12 MathSciNetzbMATHCrossRefGoogle Scholar
  76. [Uch94]
    Uchida K (1994) On Silverman’s estimate of regulators. Tohoku Math J 46:141–145 MathSciNetzbMATHCrossRefGoogle Scholar
  77. [Vor94]
    Voronoï G (1894) Concerning algebraic integers derivable from a root of an equation of the third degree. Master’s thesis, St. Petersburg Google Scholar
  78. [Was82]
    Washington LC (1982) Introduction to cyclotomic fields. GTM, vol 83. Springer, New York. 2nd edition: 1997 zbMATHCrossRefGoogle Scholar
  79. [Zim81]
    Zimmert R (1981) Ideale kleiner Norm in Idealklassen und eine Regulatorabschätzung. Invent Math 62:367–380 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2012

Authors and Affiliations

  • Olivier Bordellès
    • 1
  1. 1.AiguilheFrance

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