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Topics in Algebraic Number Theory >> Content Detail



Study Materials



Readings

Janusz = Janusz, Gerald J. Algebraic Number Fields. 2nd ed. Providence, RI: American Mathematical Society, 1996. ISBN: 0821804294.

Neukirch = Neukirch, Jürgen. Algebraische Zahlentheorie. (Algebraic Number Theory). Translated from the German by Norbert Schappacher. Berlin, Germany; New York, NY: Springer, c1999. ISBN: 3540653996.

Cassels-Fröhlich = Cassels, J. W. S., and A. Fröhlich. "Algebraic number theory." Proceedings of an instructional conference organized by the London Mathematical Society (a NATO advanced study institute) with the support of the International Mathematical Union. New York, NY: Academic Press, 1967. ISBN: 0121632512. (Out of print.)

Milne's Notes = Class Field Theory, available at James Milne's Web site.

For Class Field Theory, see also my Math 254B course notes (Berkeley, spring 2002).


Lec #TOPICSREADINGS
1Course OverviewElliptic Curves (PDF)
2Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition)Janusz, sections I.1-3.
3Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of IdealsJanusz, section I.3.
4Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, DiscriminantJanusz, sections I.4-5.
5Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable CaseJanusz, sections I.5-6.
6Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, DiscriminantJanusz, sections I.6-7.
7Discriminants and Ramification, Norms of IdealsJanusz, sections I.7-8.
8Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case)Janusz, sections I.8 and I.10.
9Linearly Disjoint Extensions; Cyclotomic Fields (General Case)Janusz, sections I.9, I.10, and I.11.

See also this supplement (PDF)
10Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, LatticesJanusz, sections I.11, I.12, and I.13.
11Lattices and Ideal Classes, Minkowski's Theorem, Finiteness of the Class Group; Dirichlet's Units TheoremJanusz, sections I.12 and I.13.
12Proof of Dirichlet's Units TheoremJanusz, section I.13.
13Absolute Values; Completions of Fields with Respect to an Absolute Value, Examples; Dichotomy between Archimedean Nonarchimedean Absolute Values; Absolute Values Coming from Discrete Valuation Rings; Normalized Absolute Values (Places), Statement of the Product Formula for Number Fields; Classification of Completions of the Rational Numbers (Ostrowski's Theorem)Janusz, sections II.1-II.3.
14In-class Midterm Exam
15Ostrowski's Theorem (cont.); Exponential and Logarithm Series; Hensel's Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute ValuesJanusz, sections II.2 and II.3.
16Extension of Nonarchimedean Absolute ValuesJanusz, section II.3.
17Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified ExtensionsJanusz, sections II.3 and II.5.
18Decomposition and Inertia Groups, Frobenius Elements, Artin SymbolsJanusz, sections III.1 and III.2.
19Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic CaseJanusz, sections III.3 and IV.1.
20More on Ray Class Groups; Idelic InterpretationJanusz, section IV.1.

Neukirch, section VI.1.
21Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet's Theorem and GeneralizationsJanusz, section IV.2.
22Chebotarev Density Theorem; Arakelov Class GroupSee the Arakelov Class Group Notes by Rene Schoof (PDF)

Janusz, section IV.3.
23Arakelov Class Group (cont.); Local Class Field TheorySee the Arakelov Class Group Notes by Rene Schoof (PDF)

Neukirch, section III.

Milne's Notes.
24Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal TheoremNeukirch, sections III and VI.

Milne's Notes.

Cassels-Fröhlich.
25Class Field Towers; Complex MultiplicationCassels-Fröhlich.

 








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