Contents

Idea

Algebraic number theory studies algebraic numbers, number fields and related algebraic structures.

An algebraic number is a root of a polynomial equation with integer coefficients (or, equivalently with rational coeffients). An algebraic integer is a root of a monic polynomial with integer coefficients. Given a field $k$ a (algebraic) number field $K = k[P]$ over $k$ is the minimal field containing all the roots of a given polynomial $P$ with coefficients in $k$. Usually one considers algebraic number fields over rational numbers.

The main direction in algebraic number theory is the class field theory which roughly studies finite abelian extensions of number fields. The one dimensional class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse…) and are quite different for the local field from the global field case. Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Tate, Kato, Saito etc.).

Related entries

• field, characteristic, division ring, ideal

• integer, rational number, irrational number, period

• p-adic number, profinite group, idele, adele

• separable extension, normal extension of fields?, Galois group, Galois extension, abelian extension of fields?, cyclotomic field

• Brower group?, Galois cohomology

• Dedekind ring, principal ideal ring?, unique factorization domain, integral closure

• algebraic number, algebraic closure, number field

• discriminant, resultant, Euclid algorithm?

• Diophantine equation, Matiyasevich theorem?

• function field

• discrete valuation, valuation ring, valuation ideal?, archimedean valuation

• local field, global field, complete field

• conductor, ideal class group, Picard group, Milnor K-group?

• class field theory, global class field theory?, local class field theory?, higher class field theory?, Hilbert class field?

• reciprocity law, Artin reciprocity law, Weil reciprocity law

• arithmetic scheme, Arakelov geometry, field with one element

• L-function, motive, Riemann conjecture, algebraic K-theory, Grothendieck Galois theory

Literature

• Albrecht Fröhlich, J. W. S. Cassels (eds.), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, pdf, errata pdf by Kevin Buzzard)

• J. W. S. Cassels, Local Fields, Cambridge University Press, 1986 (ISBN:9781139171885, doi:10.1017/CBO9781139171885)

• Jürgen Neukirch, Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)

• Albrecht Fröhlich, Martin J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics 27 (1993)

• Serge Lang, Algebraic number theory, Graduate Texts in Mathematics 110, Springer (1970, 1986, 1994, 2000) [doi:10.1007/978-1-4612-0853-2]

• James Milne, Algebraic number theory (2020) [pdf]

The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory

• P. Almeida, Noncommutative geometry and arithmetics, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, doi, see also nLab:arithmetic and noncommutative geometry

See also

• Alexander Schmidt, Higher dimensional class field theory from a topological point of view, page