18.786
Number Theory II
Continuation of 18.785 . More advanced topics in number theory, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, or quadratic forms.
Spring 2016 · Number Theory II: Class Field Theory
This course is the continuation of [_18.785 Number Theory I_](/courses/18-785-number-theory-i-fall-2019/). It begins with an analysis of the quadratic case of Class Field Theory via Hilbert symbols, in order to give a more hands-on introduction to the ideas of Class Field Theory. More advanced topics in number theory are discussed in this course, such as Galois cohomology, proofs of class field theory, modular forms and automorphic forms, Galois representations, and quadratic forms.
Readings
- Serre, Jean-Pierre. A Course in Arithmetic . New York: Springer, 2001. ISBN: 9780387900407. The Hilbert symbol is the main subject of Part I of this book, especially Chapter III. It only treats completions of Q, but the material generalizes away from Q2 to general local fields. The book emphasizes elementary techniques, so can give a feeling for what is going on in a more abstract framework.
- Serre, Jean-Pierre. Local Fields . Vol. 67. New York, NY: Springer, 2013. ISBN: 9781475756739. A classic reference that rewards the effort you put into it. It begins with the structure theory of local fields, develops group cohomology from scratch, and then proves the main theorem of local class field theory. Unfortunately, this book does not do the work of plainly laying bare its main threads, so requires some patience for self-study.
- Artin, Emil, and John Torrence Tate. Class Field Theory . Vol. 366. American Mathematical Society, 1967. An original source for many of the ideas of global class field theory. Unfortunately, it does not treat local class field theory.
- 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: 9780121632519. Notes from an instructional conference, developing the whole theory more or less from scratch. Notes available from many different authors. The quality sometimes varies, but is often high. In particular, Serre and Tate contribute the notes on local and global class field theory respectively, and generally speaking, anything written by either of them is required reading.
- Weil, André. Basic Number Theory . Vol. 144. New York, NY: Springer, 2013. ISBN: 9781461298366. Another classic text. It gives a cohomological treatment of class field theory without every saying the words, which is both a bug and a feature. It is the only source I know with a detailed approach to the proof of the main global theorem via zeta functions.
- Neukirch, Jürgen. Class Field Theory . Berlin, Heidelberg: Springer Berlin Heidelberg, 1986.
- Neukirch, Jürgen. Class Field Theory: The Bonn Lectures . Heidelberg: Springer, 2013. ISBN: 9783642354366. The former is really a geodesic approach to the subject that minimizes the role of group cohomology, only using it in the case of cyclic groups, where it is more elementary.
- Neukirch, Jürgen. Algebraic Number Theory . Vol. 322. Berlin: Springer, 2010. ISBN: 9783642084737. This is a great introduction for the general background on number fields.
- Neukirch, Jürgen, Alexander Schmidt, and Kay Wingberg. Cohomology of Number Fields . 2013. ISBN: 9783540378884. This is a more advanced treatment of Galois cohomology and its role in arithmetic.
- Riemann, B. “On the Number of Primes Less Than a Given Magnitude.” Monthly Reports of the Berlin Academy . 1859. This is still the classic reference and definitely worth a read. English translations are readily available.
- Tate, John Torrence. “Fourier Analysis in Number Fields and Hecke’s Zeta-functions.” PhD diss., Princeton University, Princeton, NJ, 1950. Tate’s thesis, which was reprinted in Cassels-Froöhlich. Its major contribution is to reinterpret Riemann’s work on the analytic properties of the zeta function by using Fourier analysis not on R/Z, but on AQ/Q. This is very useful for generalizing to number fields (c.f. to the treatment 2 of Hecke’s work in Neukirch’s Algebraic number theory), and much more clearly highlights the mechanisms underlying the analytic theory.
- Gelfand, Sergei I. and Yuri I. Manin, Methods of Homological Algebra . Berlin: Springer, 2011. ISBN: 9783642078132. A great place to learn the subject. Different people tend to take different things away from it, which is a great sign of its richness. It gives a heavy emphasis on triangulated categories, and I personally think it would be improved by focusing more on chain complexes themselves.
- Grothendieck, Alexandre. “Sur Quelques Points D’algèbre Homologique.” Tohoku Mathematical Journal , 2nd series, vol. 9, 1957: 119–183. There’s a lot that’s great about this paper still, but also a lot that’s dated. For example, I find the ∂-functors story to be dated (and uninspiring).
- Cartan, Henri, and Samuel Eilenberg. Homological Algebra . Princeton (New Jersey): Princenton University Press, 1999. ISBN: 9780691049915. Another old text, with similar deficiencies. But it has many virtues, and is filled with examples. Much of the subject was invented here, which also makes it a great source from which to learn.
Spring 2010 · Topics in Algebraic Number Theory
This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.
Readings
- Samuel, Pierre. Algebraic Theory of Numbers . Translated by Allan J. Silberger. Mineola, NY: Dover, 2008. ISBN: 9780486466668.
Spring 2006 · Topics in Algebraic Number Theory
Topics
- Course Overview
- Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition)
- Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals
- Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant
- Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case
- Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant
- Discriminants and Ramification, Norms of Ideals
- Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case)
- Linearly Disjoint Extensions; Cyclotomic Fields (General Case)
- Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices
- Lattices and Ideal Classes, Minkowski’s Theorem, Finiteness of the Class Group; Dirichlet’s Units Theorem
- Proof of Dirichlet’s Units Theorem
- 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)
- Ostrowski’s Theorem (cont.); Exponential and Logarithm Series; Hensel’s Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values
- Extension of Nonarchimedean Absolute Values
- Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions
- Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols
- Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case
- More on Ray Class Groups; Idelic Interpretation
- Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet’s Theorem and Generalizations
- Chebotarev Density Theorem; Arakelov Class Group
- Arakelov Class Group (cont.); Local Class Field Theory
- Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem
- Class Field Towers; Complex Multiplication
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.)
- Lec #: 2 · TOPICS: Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition) · READINGS: Janusz, sections I.1-3.
- Lec #: 3 · TOPICS: Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals · READINGS: Janusz, section I.3.
- Lec #: 4 · TOPICS: Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant · READINGS: Janusz, sections I.4-5.
- Lec #: 5 · TOPICS: Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case · READINGS: Janusz, sections I.5-6.
- Lec #: 6 · TOPICS: Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant · READINGS: Janusz, sections I.6-7.
- Lec #: 7 · TOPICS: Discriminants and Ramification, Norms of Ideals · READINGS: Janusz, sections I.7-8.
- Lec #: 8 · TOPICS: Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case) · READINGS: Janusz, sections I.8 and I.10.
- Lec #: 10 · TOPICS: Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices · READINGS: Janusz, sections I.11, I.12, and I.13.
- Lec #: 11 · TOPICS: Lattices and Ideal Classes, Minkowski’s Theorem, Finiteness of the Class Group; Dirichlet’s Units Theorem · READINGS: Janusz, sections I.12 and I.13.
- Lec #: 12 · TOPICS: Proof of Dirichlet’s Units Theorem · READINGS: Janusz, section I.13.
- Lec #: 13 · TOPICS: 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) · READINGS: Janusz, sections II.1-II.3.
- Lec #: 15 · TOPICS: Ostrowski’s Theorem (cont.); Exponential and Logarithm Series; Hensel’s Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values · READINGS: Janusz, sections II.2 and II.3.
- Lec #: 16 · TOPICS: Extension of Nonarchimedean Absolute Values · READINGS: Janusz, section II.3.
- Lec #: 17 · TOPICS: Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions · READINGS: Janusz, sections II.3 and II.5.
- Lec #: 18 · TOPICS: Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols · READINGS: Janusz, sections III.1 and III.2.
- Lec #: 19 · TOPICS: Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case · READINGS: Janusz, sections III.3 and IV.1.
- Lec #: 21 · TOPICS: Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet’s Theorem and Generalizations · READINGS: Janusz, section IV.2.
- Lec #: 25 · TOPICS: Class Field Towers; Complex Multiplication · READINGS: Cassels-Fröhlich.
Prerequisites & course information
Prereq: 18.785 G (Spring) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2016
- MIT OCW readings ↗
- MIT OCW syllabus ↗
- MIT OpenCourseWare ↗Spring 2010
- MIT OCW syllabus ↗
- MIT OpenCourseWare ↗Spring 2006
- MIT OCW calendar ↗
- MIT OCW readings ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.