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This is the first semester of a one-year graduate course in number theory covering standard topics in algebraic and analytic number theory.
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English
English [CC]
FREE
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Description
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 content
- 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
Instructor
Massachusetts Institute of Technology
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