18.785 | Fall 2021 | Graduate

Number Theory I

Lecture Notes

The complete lecture notes Number Theory I (PDF - 2.7 MB) can be used as the online textbook for this course.

Lecture 1: Absolute Values and Discrete Valuations (PDF)

Lecture 2: Localization and Dedekind Domains (PDF)

Lecture 3: Properties of Dedekind Domains and Factorization of Ideals (PDF)

Lecture 4: Étale Algebras, Norm and Trace (PDF)

Lecture 5: Dedekind Extensions (PDF)

Lecture 6: Ideal Norms and the Dedekind-Kummer Theorem (PDF)

Lecture 7: Galois Extensions, Frobenius Elements, and the Artin Map (PDF)

Lecture 8: Complete Fields and Valuation Rings (PDF)

Lecture 9: Local Fields and Hensel’s Lemmas (PDF)

Lecture 10: Extensions of Complete DVRs (PDF)

Lecture 11: Totally Ramified Extensions and Krasner’s Lemma (PDF)

Lecture 12: The Different and the Discriminant (PDF)

Lecture 13: Global Fields and the Product Formula (PDF)

Lecture 14: The Geometry of Numbers (PDF)

Lecture 15: Dirichlet’s Unit Theorem (PDF)

Lecture 16: Riemann’s Zeta Function and the Prime Number Theorem (PDF)

Lecture 17: The Functional Equation (PDF)

Lecture 18: Dirichlet L-functions and Primes in Arithmetic Progressions (PDF)

Lecture 19: The Analytic Class Number Formula (PDF)

Lecture 20: The Kronecker-Weber Theorem (PDF)

Lecture 21: Class Field Theory: Ray Class Groups and Ray Class Fields (PDF)

Lecture 22: The Main Theorems of Global Class Field Theory (PDF)

Lecture 23: Tate Cohomology (PDF)

Lecture 24: Artin Reciprocity in the Unramified Case (PDF)

Lecture 25: The Ring of Adeles and Strong Approximation (PDF)

Lecture 26: The Idele Group, Profinite Groups, and Infinite Galois Theory (PDF)

Lecture 27: Local Class Field Theory (PDF)

Lecture 28: Global Class Field Theory and the Chebotarev Density Theorem (PDF)

Course Info

As Taught In
Fall 2021
Learning Resource Types
Online Textbook
Lecture Notes
Problem Sets