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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11R: Algebraic number theory: global fields

Introduction

History

Applications and related fields

For complex multiplication, See 11G15

The rings of integers in these fields are also studied in Commutative Ring Theory, where we may find discussion of Unique Factorization Domains (UFDs), Euclidean domains, etc.

Subfields

• 11R04: Algebraic numbers; rings of algebraic integers
• 11R06: PV-numbers and generalizations; other special algebraic numbers
• 11R09: Polynomials (irreducibility, etc.)
• 11R11: Quadratic extensions
• 11R16: Cubic and quartic extensions
• 11R18: Cyclotomic extensions
• 11R20: Other abelian and metabelian extensions
• 11R21: Other number fields
• 11R23: Iwasawa theory
• 11R27: Units and factorization
• 11R29: Class numbers, class groups, discriminants
• 11R32: Galois theory
• 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers, See Also 20C10
• 11R34: Galois cohomology, See also 12Gxx, 16H05, 19A31
• 11R37: Class field theory
• 11R39: Langlands-Weil conjectures, nonabelian class field theory, See also 11Fxx, 22E55
• 11R42: Zeta functions and L-functions of number fields, See also 11M41, 19F27
• 11R44: Distribution of prime ideals, See also 11N05
• 11R45: Density theorems
• 11R47: Other analytic theory, See also 11Nxx
• 11R52: Quaternion and other division algebras: arithmetic, zeta functions
• 11R54: Other algebras and orders, and their zeta and L-functions, See also 11S45, 16H05, 16Kxx
• 11R56: Adele rings and groups
• 11R58: Arithmetic theory of algebraic function fields, See also 14-XX
• 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.) [new in 2000]
• 11R65: Class groups and Picard groups of orders
• 11R70: K-theory of global fields, See also 19Fxx
• 11R80: Totally real and totally positive fields, See also 12J15
• 11R99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

There are quite a few texts in algebraic number theory. We mention a few with widely varying foci:

• Hasse, Helmut: "Number theory",Grundlehren der Mathematischen Wissenschaften 229, Springer-Verlag, Berlin-New York, 1980. 638 pp. ISBN 3-540-08275-1 (A classic reference)
• Weil, André: "Basic number theory", Classics in Mathematics, Springer-Verlag, Berlin, 1995. 315 pp. ISBN 3-540-58655-5
• Artin, Emil; Tate, John: "Class field theory", Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. 259 pp. ISBN 0-201-51011-1
• Lang, Serge: "Algebraic number theory", Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994. 357 pp. ISBN 0-387-94225-4
• Serre, Jean-Pierre: "Galois cohomology", Springer-Verlag, Berlin, 1997. 210 pp. ISBN 3-540-61990-9
• Borevich, A. I.; Shafarevich, I. R.: "Number theory", Academic Press, New York-London 1966 435 pp.
• Janusz, Gerald J.: "Algebraic number fields", American Mathematical Society, Providence, RI, 1996. 276 pp. ISBN 0-8218-0429-4
• Pohst, M.; Zassenhaus, H.: "Algorithmic algebraic number theory", Encyclopedia of Mathematics and its Applications, 30. Cambridge University Press, Cambridge, 1989. 465 pp. ISBN 0-521-33060-2

Possibly useful survey articles include:

• Mines, Ray, "Algebraic number theory, a survey", The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981), 337--358, Stud. Logic Found. Math., 110. North-Holland, Amsterdam-New York, 1982.
• Gelbart, Stephen: "An elementary introduction to the Langlands program", Bull. Amer. Math. Soc. 10 (1984) 177--219.
• Garbanati, Dennis, "Class field theory summarized", Rocky Mountain J. Math. 11 (1981), no. 2, 195--225. MR 82g:12010

There is also a mailing list "ALGEBRAIC-NUMBER-THEORY"; information is available at the Algebraic Number Theory Preprint Archive

See also the references for number theory in general.

Software and tables

• Announcement: Table of number fields (Henri Cohen)

Other web sites with this focus

Selected topics at this site

• The numbers sin((p/q)*Pi) are algebraic.
• Linear relations between roots of unity
• How to decide if an algebraic number is a square (in which ring?...)
• Products of prime, or irreducible, elements of a domain
• Unusual consequence of unique factorization
• Identification of fundamental unit in a real quadratic number field
• Abhyankhar's Lemma (ramification of primes in compositum of extensions)
• Some nested radical expressions simplify
• Class field theory in case of dihedral Galois groups
• Predicting the Galois group of a polynomial using Cebotarev's Theorem (f splits mod p for 1/|G| of all primes p)
• What are the quadratic fields with small class numbers? (literature citations)
• Compute class numbers by counting inequivalent quadratic forms.
• What is a class number? (e.g. counting equivalence classes of quadratic forms)
• Pairing off ideal classes with classes of quadratic forms
• Which cyclotomic fields are unique factorization domains? (What is their class number?)
• Relationship between unique factorization domains and the near-integrality of exp(pi sqrt(163))
• What is the correct sign in the congruence ((p-1)/2)! = +-1 mod p? (cf. Wilson's theorem). Answer: depends on class number formula.
• Finding nice (small) generators for the integers in a number field.
• [Offsite] What is Class Field Theory [Tim Chow]
• Enumerating imaginary quadratic fields having a given small class number
• The Hasse Principle: a quadratic equation may be solved rationally if it may be solved mod p for every p (with or without solvability in the reals).
• References for the local-to-global principle for quadratic equations (Brauer-Hasse-Noether theorem for associative algebras).
• Examples of failure of local-to-global principle.
• An elliptic curve formulation of the n = 3 case of Fermat's Last Theorem; in which quadratic extensions of Q does it have a solution?
• What is the Langlands Program?
• What is the Langlands program?
• Explicit Cebotarev density: which primes split well in algebraic number fields?
• Are there algebraic numbers on the unit circle besides roots of unity? (yes, many)
• Why are there so many primes of the form n^2+n+41? (Citations)
• What's the connection between unique factorization and the preponderance of primes of the form x^2-x+41?
• What is a regular prime?