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November 7, 2022


Unlimited Duration


This course includes:

Unlimited Duration

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Certificate of completion

Unlimited Duration


This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory.

At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

Course Curriculum

  • Absolute Values and Discrete Valuations Unlimited
  • Localization and Dedekind Domains Unlimited
  • Properties of Dedekind Domains and Factorization of Ideals Unlimited
  • Étale Algebras, Norm and Trace Unlimited
  • Dedekind Extensions Unlimited
  • Ideal Norms and the Dedekind-Kummer Theorem Unlimited
  • Galois Extensions, Frobenius Elements, and the Artin Map Unlimited
  • Complete Fields and Valuation Rings Unlimited
  • Local Fields and Hensel’s Lemmas Unlimited
  • Extensions of Complete DVRs Unlimited
  • Totally Ramified Extensions and Krasner’s Lemma Unlimited
  • The Different and the Discriminant Unlimited
  • Global Fields and the Product Formula Unlimited
  • The Geometry of Numbers Unlimited
  • Dirichlet’s Unit Theorem Unlimited
  • Riemann’s Zeta Function and the Prime Number Theorem Unlimited
  • The Functional Equation Unlimited
  • Dirichlet L-functions and Primes in Arithmetic Progressions Unlimited
  • The Analytic Class Number Formula Unlimited
  • The Kronecker-Weber Theorem Unlimited
  • Class Field Theory: Ray Class Groups and Ray Class Fields Unlimited
  • The Main Theorems of Global Class Field Theory Unlimited
  • Tate Cohomology Unlimited
  • Artin Reciprocity in the Unramified Case Unlimited
  • The Ring of Adeles and Strong Approximation Unlimited
  • The Idele Group, Profinite Groups, and Infinite Galois Theory Unlimited
  • Local Class Field Theory Unlimited
  • Global Class Field Theory and the Chebotarev Density Theorem Unlimited

About the instructor

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Massachusetts Institute of Technology