Number theory
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Number Theory — scope drawn from open graduate number theory notes and typical US graduate number theory syllabi.
Algebraic number theory
- Dedekind domains and fractional ideals — Dedekind domains and fractional ideals
- Class groups — Class groups
- Finiteness of the class number — Finiteness of the class number
- Ramification — Ramification
- Splitting of primes in extensions — Splitting of primes in extensions
- Discriminant and integral bases — Discriminant and integral bases
- Minkowski's lattice point theorem — Minkowski's lattice point theorem
- Applications — Applications
Local fields and adeles (introduction)
- p-adic numbers and valuations — p-adic numbers and valuations
- Completion of Q with respect to the p-adic valuation — Completion of Q with respect to the p-adic valuation
- Local class field theory (overview) — Local class field theory (overview)
- Adeles and ideles (introduction) — Adeles and ideles (introduction)
- Chevalley's product formula (statement) — Chevalley's product formula (statement)
Analytic methods
- Dirichlet series and the Riemann zeta function — Dirichlet series and the Riemann zeta function
- Prime number theorem (proof outline) — Prime number theorem (proof outline)
- L-functions — L-functions
- Dirichlet's theorem on primes in arithmetic progressions — Dirichlet's theorem on primes in arithmetic progressions
- Modular forms (introduction) — Modular forms (introduction)
- Elliptic curves: Mordell–Weil theorem (statement) — Elliptic curves: Mordell–Weil theorem (statement)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Algebraic number theory
Dedekind domains and fractional ideals
Coming soon - Local fields and adeles (introduction)
p-adic numbers and valuations
Coming soon - Analytic methods
Dirichlet series and the Riemann zeta function
Coming soon
Notes
Graduate Number Theory — scope drawn from open graduate number theory notes and typical US graduate number theory syllabi. Topic outline: `content/topics/graduate/number_theory.json`.