User:RobHar/Sandbox3

Basic notions

• Algebraic number fields
• Class groups
• Primes and places
• Unit groups
• Discriminant of an algebraic number field
• Conductor
• Localization

Branches of algebraic number theory

• Class field theory
• Iwasawa theory
• Galois representations
• Arithmetic geometry

Major results

• Finiteness of the class number
• Dirichlet's unit theorem
• Kronecker–Weber theorem
• Artin reciprocity
• Class number formula

Open questions

• Non-abelian class field theory
• Special values of L-functions

• Birch-Swinnerton-Dyer
• Bloch-Kato
• Iwasawa "main conjectures"

References

Graduate level accounts

• Cassels, J. W. S.; Frölich, Albrecht, eds. (1967), Algebraic number theory, London: Academic Press, MR 0215665
• Fröhlich, Albrecht; Taylor, Martin J. (1993), Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, ISBN 0-521-43834-9, MR 1215934
• Lang, Serge (1994), Algebraic number theory, Graduate Texts in Mathematics, vol. 110 (2 ed.), New York: Springer-Verlag, ISBN 0-387-94225-4, MR 1282723
• Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.