Theoretical / proof-based
Number theory · Graduate · Math
Learning objectives from the Number theory syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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)
Definitions and structure
- 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
Proofs and reasoning
- Discriminant and integral bases — Discriminant and integral bases
- Minkowski's lattice point theorem — Minkowski's lattice point theorem
- Applications — Applications
- 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
Abstraction and generalization
- 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)
- Dirichlet series and the Riemann zeta function — Dirichlet series and the Riemann zeta function
- Prime number theorem (proof outline) — Prime number theorem (proof outline)
Modeling and computation
- Apply dedekind domains and fractional ideals in engineering contexts — Apply dedekind domains and fractional ideals in engineering contexts
- Apply class groups in engineering contexts — Apply class groups in engineering contexts
- Apply finiteness of the class number in engineering contexts — Apply finiteness of the class number in engineering contexts
- Apply ramification in engineering contexts — Apply ramification in engineering contexts
- Apply splitting of primes in extensions in engineering contexts — Apply splitting of primes in extensions in engineering contexts
Data and technology
- Use software to explore number theory problems numerically — Use software to explore number theory problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Discriminant and integral bases — Discriminant and integral bases
- Minkowski's lattice point theorem — Minkowski's lattice point theorem
- Applications — Applications
- 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
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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