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18.785

Number Theory I

3-0-9 units · Graduate

Dedekind domains, unique factorization of ideals, splitting of primes. Lattice methods, finiteness of the class group, Dirichlet's unit theorem. Local fields, ramification, discriminants. Zeta and L-functions, analytic class number formula. Adeles and ideles. Statements of class field theory and the Chebotarev density theorem.

Fall 2021

Topics

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

Readings

Prerequisites & course information

Prereq: None. Coreq: 18.705 G (Fall) 3-0-9 units

Sources