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Browse materialsMIT 18.786 · Spring 2016

Number Theory II: Class Field Theory

Subjects: Algebra · Geometry · Mathematics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor: Dr. Sam Raskin
Term: Spring 2016
Level: Graduate
What's inside: 7 problem sets · 2 solutions · 1 lecture note · 1 reading · 24 other files · 2 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.786 Number Theory II: Class Field Theory, Spring 2016, Dr. Sam Raskin — 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.

Has MIT solutions 2 files in this course are named by MIT as solutions.

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

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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.

Lecture Notes

24 files

Assignments

9 problem sets

MIT OpenCourseWare, Dr. Sam Raskin, 18.786 Number Theory II: Class Field Theory, Spring 2016.

License: CC BY-NC-SA 4.0 · Edition: Spring 2016 · Open the original course on OCW ↗

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Number Theory II: Class Field Theory

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Lecture 1: Introduction (PDF)

Lecture 2: Hilbert Symbols (PDF)

Lecture 3: Norm Groups with Tame Ramification (PDF)

Lecture 4: GCFT and Quadratic Reciprocity (PDF)

Lecture 5: Non-Degeneracy of the Adèle Pairing and Exact Sequences (PDF)

Lecture 6: Exact Sequences and Tate Cohomology (PDF)

Lecture 7: Chain Complexes and Herbrand Quotients (PDF)

Lecture 8: Tate Cohomology and Inverse Limits (PDF)

Lecture 9: Hilbert’s Theorem 90 and Cochain Complexes (PDF)

Lecture 10: Homotopy, Quasi-Isomorphism, and Coinvariants (PDF)

Lecture 11: The Mapping Complex and Projective Resolutions (PDF)

Lecture 12: Derived Functors and Explicit Projective Resolutions (PDF)

Lecture 13: Homotopy Coinvariants, Abelianization, and Tate Cohomology (PDF)

Lecture 14: Tate Cohomology and K{{< sup “unr” >}} (PDF)

Lecture 15: The Vanishing Theorem Implies Cohomological LCFT (PDF)

Lecture 16: Vanishing of Tate Cohomology Groups (PDF)

Lecture 17: Proof of the Vanishing Theorem (PDF)

Lecture 18: Norm Groups, Kummer Theory, and Profinite Cohomology (PDF)

Lecture 19: Brauer Groups (PDF)

Lecture 20: Proof of the First Inequality (PDF)

Lecture 21: Artin and Brauer Reciprocity, Part I (PDF)

Lecture 22: Artin and Brauer Reciprocity, Part II (PDF)

Lecture 23: Proof of the Second Inequality (PDF)