Free lecture videos by a Harvard professor on abstract algebra
Benedict Gross, PhD, George Vasmer Leverett Professor of Mathematics, Harvard University.
Algebra is the language of modern mathematics. This course introduces students to that language through a study of groups, group actions, vector spaces, linear algebra, and the theory of fields.
The lectures videos
The recorded lectures are from the Harvard Faculty of Arts and Sciences course Mathematics 122, which was offered as an online course at the Extension School.
The Quicktime and MP3 formats are available for download, or you can play the Flash version directly. Each week has 3 lectures that are 50 minutes each.
Review of linear algebra
Groups. Examples of groups. Basic properties and constructions.
- Video/Audio
- Introduction to the course; Review: Linear algebra; Definition of groups
- Administrative notes; Generalities on groups; Symmetric groups on n letters; A stabilizer subgroup; The subgroups of Z; Cyclic subgroups gen by element
- The story so far; Isomorphisms; Homomorphisms; Images
Permutations
Cosets, Z/nZ.
- Video/Audio
- Review, kernels, normality; Examples; Centers and inner autos
- Equivalence relations; Cosets; Examples
- Congruence mod n; (Z/nZ)*
Quotient groups, first isomorphism theorem
Abstract fields, abstract vectorspaces. Construction and invariants of vectorspaces.
- Video/Audio
- Quotients
- More on quotients; Vectorspaces
- Continued
Abstract linear operators and how to calculate with them
Properties and construction of operators.
- Video/Audio
- Bases and vectorspaces; Matrices and linear transfs
- Bases; Matrices
- Eigenvalues and eigenvectors
- Review for midterm; Orthogonal group
Orthogonal groups
- Video/Audio
- Orthogonal group & geometry
- Finite groups of motions
- Discrete groups of motions
Isometrics of plane figures
Cyclic and dihedral groups. Finite and discrete subgroups of symmetry groups.
- Video/Audio
- Discrete groups of motions; Abstract group actions
- Group actions
- Continued
Group actions
Basic properties and constructions. Groups acting on themselves by left multiplication. Groups acting on themselves by conjugation.
- Video/Audio
- Part 1
- Part 2
- Part 3
A5 and the symmetries of an icosahedron
Sylow theorems. Study of permutation groups.
- Video/Audio
- Alternating group structure
- Rings
- Continued
Rings
Examples of rings. Basic properties and constructions.
- Video/Audio
- Part 1
- Part 2
- Part 3
Extensions of rings
Quotient rings. Integral domains, fields of fractions.
- Video/Audio
Special lecture
- Video/Audio
- Part 1
- Part 2
- Part 3
Euclidean domains, PIDs, UFDs
Gauss’ lemma. Eisenstein’s criterion. Algebraic integers.
- Video/Audio
- Part 1
- Part 2
- Part 3
Structure of ring of integers in a quadratic field
Dedekind domains. Ideal class groups.
- Video/Audio
- Part 1
Wrap-up
- Video/Audio
- Part 1
- Part 2
Class Materials
- Syllabus
- Notes
- Notes 1
- Notes 2
- Notes 3
- Notes 4
- Notes 5
- Notes 6
- Notes 7
- Notes 8
- Notes 9
- Notes 10
- Notes 11
- Notes 12
- Notes 13
- Notes 14
- Notes 15
- Notes 16
- Notes 17
- Notes 18
- Notes 19
- Notes 20
- Notes 21
- Notes 22
- Notes 23
- Notes 24
- Notes 25
- Notes 26
- Notes 27
- Notes 28
- Notes 29
- Notes 30
- Notes 31
- Notes 32
- Notes 33
- Notes 34
- Notes 35
- Notes 36
- Problem Sets
- Problem Set 1
- Problem Set 2
- Problem Set 3
- Problem Set 4
- Problem Set 5
- Problem Set 6
- Problem Set 7
- Problem Set 8
- Problem Set 9
- Problem Set 10
- Problem Set 11
- Problem Set 12
- Problem Set 13
- Problem Set 14
- Problem Set 15
- Problem Set 16
- Problem Set 17
- Problem Set 18
- Problem Set 19
- Problem Set 20
- Problem Set 21
- Problem Set 22
- Problem Set 23
- Problem Set 24
- Problem Set 25
- Problem Set 26
- Problem Set 27
- Problem Set 28
- Problem Set 29
- Problem Set 30
- Problem Set 31
- Problem Set 32
- Problem Set 33
- Problem Set 34
- Problem Set 35
- Problem Set 36
Enroll in Harvard Extension School courses
If you enjoyed this free class, the Harvard Extension School offers a wide variety of courses in numerous fields. Search for classes and enroll for credit during the fall and spring registration periods.