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# 11R: Algebraic number theory: global fields

## 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.)
• 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
• 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
• 11R45: Density theorems
• 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
• 11R60: Cyclotomic function fields (class groups, Bernoulli objects, etc.) [new in 2000]
• 11R65: Class groups and Picard groups of orders
• 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