18.786 | Spring 2006 | Graduate

Topics in Algebraic Number Theory


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).

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

Janusz, sections I.9, I.10, and I.11.

See also this supplement (PDF)

10 Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices Janusz, sections I.11, I.12, and I.13.
11 Lattices and Ideal Classes, Minkowski’s Theorem, Finiteness of the Class Group; Dirichlet’s Units Theorem Janusz, sections I.12 and I.13.
12 Proof of Dirichlet’s Units Theorem Janusz, section I.13.
13 Absolute 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.
14 In-class Midterm Exam  
15 Ostrowski’s Theorem (cont.); Exponential and Logarithm Series; Hensel’s Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values Janusz, sections II.2 and II.3.
16 Extension of Nonarchimedean Absolute Values Janusz, section II.3.
17 Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions Janusz, sections II.3 and II.5.
18 Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols Janusz, sections III.1 and III.2.
19 Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case Janusz, sections III.3 and IV.1.
20 More on Ray Class Groups; Idelic Interpretation

Janusz, section IV.1.

Neukirch, section VI.1.

21 Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet’s Theorem and Generalizations Janusz, section IV.2.
22 Chebotarev Density Theorem; Arakelov Class Group

See the Arakelov Class Group Notes by Rene Schoof (PDF)

Janusz, section IV.3.

23 Arakelov Class Group (cont.); Local Class Field Theory

See the Arakelov Class Group Notes by Rene Schoof (PDF)

Neukirch, section III.

Milne’s Notes.

24 Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem

Neukirch, sections III and VI.

Milne’s Notes.


25 Class Field Towers; Complex Multiplication Cassels-Fröhlich.

Course Info

As Taught In
Spring 2006
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Problem Sets