Calendar · Lecture Notes
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Topics in Algebraic Number Theory
Art: Johannes Kepler, Konjukce Saturnu a Jupiteru Originál, 1596 · Public domain
- Instructor
- Prof. Kiran Kedlaya
- Term
- Spring 2006
- Level
- Graduate
- What's inside
- 1 reading · 14 other files · 6 web pages
- Materials open
- MIT OCW
Other versions of this course
- 18.786 · Spring 2010 · Study here
- 18.786 · Spring 2006 · Study here (this version)
About this course
Source: MIT OpenCourseWare 18.786 Topics in Algebraic Number Theory, Spring 2006, Prof. Kiran Kedlaya — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.
Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.
Before you start
Prerequisites are not listed here. Check MIT's own syllabus for this course.
How this course was organized
What's here, and what isn't
- A lecture sequence of 25 items is confirmed. Assignment and exam materials exist too, but MIT's own order for them could not be verified, so they're shown as materials below rather than a sequence.
Not included here
9 items from the source could not be safely included:
- MIT's page format for this item wasn't recognized clearly enough to place it safely. (8 items)
- Referenced in a table, but without MIT's own verified order, so it isn't used to claim a sequence.
Materials by section
Grouped by section, as MIT organized them. Not sequenced or ordered.
Readings
Assignments
9 problem sets
Exams
Other materials from this course
Course outline
Calendar · Lecture Notes
Lecture 2: Localization, Examples; Integral Dependence, Integral Closure; Discrete Valuation Rings (Definition)
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Lecture 3: Discrete Valuation Rings (Properties), Dedekind Domains, Unique Factorization of Ideals
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Lecture 4: Fractional Ideals of a Dedekind Domain, Class Group, Finite Extensions of Fields, Norm, Trace, Discriminant
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Lecture 5: Trace and Norm, Separability, Nondegeneracy of the Trace Pairing for a Separable Extension, Extension of Dedekind Domains in the Separable Case
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Lecture 6: Extension of Prime Ideals, Relative Degree, Ramification Degree, The Fundamental Equality, Discriminant
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Lecture 7: Discriminants and Ramification, Norms of Ideals
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Lecture 8: Norm of a Prime Ideal; Properties of Cyclotomic Fields (Prime Power Case)
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Lecture 9: Linearly Disjoint Extensions; Cyclotomic Fields (General Case)
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Lecture 10: Why Quadratic Reciprocity is Now Easy; Real and Complex Embeddings, Lattices
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Lecture 11: Lattices and Ideal Classes, Minkowski’s Theorem, Finiteness of the Class Group; Dirichlet’s Units Theorem
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Lecture 12: Proof of Dirichlet’s Units Theorem
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Lecture 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)
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Lecture 14: In-class Midterm Exam
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Lecture 15: Ostrowski’s Theorem (cont.); Exponential and Logarithm Series; Hensel’s Lemma for Nonarchimedean Absolute Values; Extensions of Nonarchimedean Absolute Values
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Lecture 16: Extension of Nonarchimedean Absolute Values
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Lecture 17: Classification of Absolute Values on a Number Field; Product Formula for Number Fields; Unramified Extensions
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Lecture 18: Decomposition and Inertia Groups, Frobenius Elements, Artin Symbols
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Lecture 19: Artin Maps for Abelian Extensions; Ray Class Groups; The Artin Reciprocity Law; Proof in the Cyclotomic Case
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Lecture 20: More on Ray Class Groups; Idelic Interpretation
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Lecture 21: Dirichlet Series, Dedekind Zeta Functions, L-series, Dirichlet’s Theorem and Generalizations
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Lecture 22: Chebotarev Density Theorem; Arakelov Class Group
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Lecture 23: Arakelov Class Group (cont.); Local Class Field Theory
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Lecture 24: Local Class Field Theory (cont.); The Adelic Reciprocity Map; The Principal Ideal Theorem
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Lecture 25: Class Field Towers; Complex Multiplication
MIT OpenCourseWare, Prof. Kiran Kedlaya, 18.786 Topics in Algebraic Number Theory, Spring 2006.
License: CC BY-NC-SA 4.0 · Edition: Spring 2006 · Open the original course on OCW ↗
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