MATH 842: Algebraic number theory

Instructor: Nils Bruin
SC K 10507
(778) 782 3794
Outline: Algebraic number theory comprises the study of algebraic numbers: numbers that satisfy polynomial equations with rational coefficients. The parallels with usual integer arithmetic are striking, as are the notable differences (as, for instance, failure of unique factorization into prime factors). The subject is fundamental to any further study in number theory or algebraic geometry.

In this course we develop the tools to properly understand unique factorization and its failure. We establish fundamental results such as Dirichlet's Unit theorem and the finiteness of the ideal class group. We highlight the applicability of the algebraic tools we develop to both algebraic numbers and to algebraic curves. Depending on time and interests of the participants, we will also look into various applications and more advanced topics.

Possible topics:

  • Dedekind domains: ideals, fractional ideals, ideal factorization, and ideal class groups
  • Lattices and Minkowski Theory.
  • Dirichlet's Unit theorem
  • Finiteness of class groups
  • Ramification theory; interaction with Galois groups
  • Cyclotomic fields
  • Completions and valuations
  • Adelic approaches
  • Absolute Galois groups; profinite groups
  • Group cohomology; Galois cohomology

Course text: Milne, J.S.,
Algebraic Number Theory.
available from: (local copy)

Recommended reading:
  • Quite algebraic and very careful exposition, including lots of detail in proofs:
    Ribenboim, Paulo,
    Classical theory of algebraic numbers.

    Universitext. Springer-Verlag, New York, 2001. xxiv+681 pp. ISBN: 0-387-95070-2
  • Very compact in its exposition, but still complete. Probably one of the slickest presentations available:
    Neukirch, Jürgen,
    Algebraic number theory.
    Grundlehren der Mathematischen Wissenschaften 322. Springer-Verlag, Berlin, 1999. xviii+571 pp. ISBN: 3-540-65399-6
  • Quite complete in its coverage of theory but also paying attention to the computational side of things:
    Stein, William
    Algebraic Number Theory, a Computational Approach.
    available from: (local copy)
Lectures: Wednesdays, Fridays 14:30 - 16:20 in WMC 2501
Some Mondays (in Bold) 15:30 - 17:20 in WMC 2830
First lecture: Wednesday January 15
Exam: April 14, 9:00 - 17:00 (take-home final)
Accommodation will be made for clashes with Prel/Comp exams
Preliminary scheme:
Assignments: 30 %
Presentation: 15 %
Exam: 55 %
Lecture schedule: PRELIMINARY; we may not need all booked Mondays.
References to Milne are in terms of Definition/Remark/Theorem etc. numbers.
Date Subject References
Jan. 15 Cancelled due to snow
Jan. 17 Introduction: rings of integers [Milne 2.1-3]
Jan. 20 First properties of integral closures: Dedekind's characterization of integrality via modules; UFDs are integrally closed [Milne 2.4-14]
Jan. 22 Trace form; Discriminants [Milne 2.15-2.23]
Jan. 24 Finite generation of integer rings; how to compute them [Milne 2.24-43], we did some extra things too. For Assignment 0:Convergent Calculator
Jan. 27 Discrete valuation rings and localization [Milne 3.1-3.2; 1.10-1.13]
Jan. 29 Unique ideal factorization in Dedekind domains [Milne 3.3-3.18]
Jan. 31 The ideal class group [Milne 3.19-3.30]
Feb. 5 Ideal factorization in extensions [Milne 3.33-3.34]
Feb. 7 Ramification [Milne 3.35-3.53
Feb. 12
No Lecture
Feb. 14
No Lecture
Reading break
Feb. 24 Norms of ideals; Minkowski bound [Milne 4.1-4.10
Feb. 26 Lattices [Milne 4.11-16], proof that a discrete subgroup of $\mathbb{R}^n$ is a lattice
Feb. 28 Minkowski's convex body theorem: The class group is finite [Milne 4.17-4.28]
Mar. 2 The unit group: finite generation [Milne 5.1-5.8]
Mar. 4 The unit rank, S-units, and computing class groups and unit groups. [Milne 5.9-5.11]
Mar. 6
No Lecture
Mar. 11 Ostrowski's Theorem [Milne 7.1-7.12]
Mar. 13 Fermat's theorem for regular primes (first case only) [Milne 6.8-6.11, plus earlier results on cyclotomic extensions
Mar. 18 Weak approximation For assignment 3: Sage code for computing valuationsi
[Milne 7.1-7.21]
Mar. 20 Completions [Milne 7.23-7.30]
Mar. 23 Hensel's Lemma; norms on vector spaces over complete fields [Milne 7.31-7.37] (Milne does not do the norms on vector spaces in sufficient generality)
Mar. 25 Newton polygons; extensions [7.38-7.46] (We didn't quite follow this approach)
Mar. 27 Ramified and unramified extensions; Krasner's Lemma [Milne 7.47-7.65]
Mar. 30 Global fields; Strong approximation; Decomposition groups [Milne 8.10-8.11, 8.18-8.19] -- strong approximation is not in Milne
Apr. 1 Kummer theory: Cohomology, Galois modules, Hilbert 90, and the absolute Galois group Not in Milne
Apr. 3
Apr. 8
Apr. 14
Final Exam 9:00 - 17:00