Theoretical / proof-based
Abstract algebra · Undergraduate · Math
Learning objectives from the Abstract 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.
Undergraduate Abstract Algebra — scope drawn from open algebra texts (e.g. Judson Abstract Algebra: Theory and Applications, AIM-approved) and typical US undergraduate algebra syllabi.
Groups
- Binary operations and equivalence relations — Binary operations and equivalence relations
- Modular arithmetic — Modular arithmetic
- Groups: definitions, examples (Z, Z_n, dihedral, symmetric groups) — Groups: definitions, examples (Z, Z_n, dihedral, symmetric groups)
- Subgroups and cyclic groups — Subgroups and cyclic groups; generators and order
- Cosets, Lagrange's theorem, and index — Cosets, Lagrange's theorem, and index
- Normal subgroups and quotient groups — Normal subgroups and quotient groups
- Group homomorphisms — Group homomorphisms
- The first isomorphism theorem — The first isomorphism theorem
- Direct products — Direct products
- Simple groups (introduction) — Simple groups (introduction)
Rings and fields
- Rings: definitions and examples (Z, polynomial rings, matrix rings) — Rings: definitions and examples (Z, polynomial rings, matrix rings)
- Integral domains, fields, and ideals — Integral domains, fields, and ideals
- Quotient rings and ring homomorphisms — Quotient rings and ring homomorphisms
- Polynomial rings — Polynomial rings; irreducibility and factorization over Q and R
- Field extensions — Field extensions
- Constructibility (introduction) — Constructibility (introduction)
Optional capstone
- Sylow theorems (optional capstone in a second semester) — Sylow theorems (optional capstone in a second semester)
Learning objectives
Click an objective for study materials.
Groups
- Binary operations and equivalence relations — Binary operations and equivalence relations
- Modular arithmetic — Modular arithmetic
- Groups: definitions, examples (Z, Z_n, dihedral, symmetric groups) — Groups: definitions, examples (Z, Z_n, dihedral, symmetric groups)
- Subgroups and cyclic groups — Subgroups and cyclic groups; generators and order
- Cosets, Lagrange's theorem, and index — Cosets, Lagrange's theorem, and index
- Normal subgroups and quotient groups — Normal subgroups and quotient groups
- Group homomorphisms — Group homomorphisms
- The first isomorphism theorem — The first isomorphism theorem
Rings and fields
- Rings: definitions and examples (Z, polynomial rings, matrix rings) — Rings: definitions and examples (Z, polynomial rings, matrix rings)
- Integral domains, fields, and ideals — Integral domains, fields, and ideals
- Quotient rings and ring homomorphisms — Quotient rings and ring homomorphisms
- Polynomial rings — Polynomial rings; irreducibility and factorization over Q and R
- Field extensions — Field extensions
- Constructibility (introduction) — Constructibility (introduction)
Optional capstone
- Sylow theorems (optional capstone in a second semester) — Sylow theorems (optional capstone in a second semester)
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
Binary operations and equivalence relations
Coming soon - Rings and fields
Rings: definitions and examples (Z, polynomial rings, matrix rings)
Coming soon - Optional capstone
Sylow theorems (optional capstone in a second semester)
Coming soon
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$1,162 · Abstract algebra · 18 tutoring hrs
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