Theoretical / proof-based
Graduate algebra · Graduate · Math
Learning objectives from the Graduate algebra 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 Algebra — scope drawn from open graduate algebra notes (e.g. Milne Group Theory / field-algebra OER) and typical first-year graduate algebra syllabi.
Groups and modules
- Review of groups and homomorphisms — Review of groups and homomorphisms
- Quotient groups — Quotient groups
- Sylow theorems and solvable groups — Sylow theorems and solvable groups
- Modules over rings — Modules over rings
- Projective modules — Projective modules
- Structure theorem for modules over PIDs (as covered) — Structure theorem for modules over PIDs (as covered)
Rings and fields
- Rings, ideals, and quotient rings — Rings, ideals, and quotient rings
- Polynomial rings and factorization — Polynomial rings and factorization
- Field extensions and algebraic closures — Field extensions and algebraic closures
- Galois theory — Galois theory
- Applications — Applications
Commutative and homological intro
- Noetherian rings — Noetherian rings
- Primary decomposition (introduction) — Primary decomposition (introduction)
- Tensor products of modules — Tensor products of modules
- Ext and Tor (overview, if in syllabus) — Ext and Tor (overview, if in syllabus)
Problem sets
- Research-level algebra problem sets — Research-level algebra problem sets
- Qualifying-exam style proof practice by topic area — Qualifying-exam style proof practice by topic area
Learning objectives
Click an objective for study materials.
Groups and modules
- Review of groups and homomorphisms — Review of groups and homomorphisms
- Quotient groups — Quotient groups
- Sylow theorems and solvable groups — Sylow theorems and solvable groups
- Modules over rings — Modules over rings
- Projective modules — Projective modules
- Structure theorem for modules over PIDs (as covered) — Structure theorem for modules over PIDs (as covered)
Rings and fields
- Rings, ideals, and quotient rings — Rings, ideals, and quotient rings
- Polynomial rings and factorization — Polynomial rings and factorization
- Field extensions and algebraic closures — Field extensions and algebraic closures
- Galois theory — Galois theory
- Applications — Applications
Commutative and homological intro
- Noetherian rings — Noetherian rings
- Primary decomposition (introduction) — Primary decomposition (introduction)
- Tensor products of modules — Tensor products of modules
- Ext and Tor (overview, if in syllabus) — Ext and Tor (overview, if in syllabus)
Problem sets
- Research-level algebra problem sets — Research-level algebra problem sets
- Qualifying-exam style proof practice by topic area — Qualifying-exam style proof practice by topic area
Definitions and structure
- Review of groups and homomorphisms — Review of groups and homomorphisms
- Quotient groups — Quotient groups
- Sylow theorems and solvable groups — Sylow theorems and solvable groups
- Modules over rings — Modules over rings
- Projective modules — Projective modules
Proofs and reasoning
- Structure theorem for modules over PIDs (as covered) — Structure theorem for modules over PIDs (as covered)
- Rings, ideals, and quotient rings — Rings, ideals, and quotient rings
- Polynomial rings and factorization — Polynomial rings and factorization
- Field extensions and algebraic closures — Field extensions and algebraic closures
- Galois theory — Galois theory
Abstraction and generalization
- Applications — Applications
- Noetherian rings — Noetherian rings
- Primary decomposition (introduction) — Primary decomposition (introduction)
- Tensor products of modules — Tensor products of modules
- Ext and Tor (overview, if in syllabus) — Ext and Tor (overview, if in syllabus)
Modeling and computation
- Apply review of groups and homomorphisms in engineering contexts — Apply review of groups and homomorphisms in engineering contexts
- Apply quotient groups in engineering contexts — Apply quotient groups in engineering contexts
- Apply sylow theorems and solvable groups in engineering contexts — Apply sylow theorems and solvable groups in engineering contexts
- Apply modules over rings in engineering contexts — Apply modules over rings in engineering contexts
- Apply projective modules in engineering contexts — Apply projective modules in engineering contexts
Data and technology
- Use software to explore graduate algebra problems numerically — Use software to explore graduate algebra 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
- Structure theorem for modules over PIDs (as covered) — Structure theorem for modules over PIDs (as covered)
- Rings, ideals, and quotient rings — Rings, ideals, and quotient rings
- Polynomial rings and factorization — Polynomial rings and factorization
- Field extensions and algebraic closures — Field extensions and algebraic closures
- Galois theory — Galois theory
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.
- Groups and modules
Review of groups and homomorphisms
Coming soon - Rings and fields
Rings, ideals, and quotient rings
Coming soon - Commutative and homological intro
Noetherian rings
Coming soon - Problem sets
Research-level algebra problem sets
Coming soon
Pricing calculator
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$1,162 · Graduate algebra · 18 tutoring hrs
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