#### Study at Harvard this summer

Harvard Summer School courses are open for registration through May 19.

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 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.

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

Cosets, Z/nZ.

- Video/Audio
- Review, kernels, normality; Examples; Centers and inner autos
- Equivalence relations; Cosets; Examples
- Congruence mod n; (Z/nZ)*

Abstract fields, abstract vectorspaces. Construction and invariants of vectorspaces.

- Video/Audio
- Quotients
- More on quotients; Vectorspaces
- Continued

Properties and construction of operators.

- Video/Audio
- Bases and vectorspaces; Matrices and linear transfs
- Bases; Matrices
- Eigenvalues and eigenvectors
- Review for midterm; Orthogonal group

- Video/Audio
- Orthogonal group & geometry
- Finite groups of motions
- Discrete groups of motions

Cyclic and dihedral groups. Finite and discrete subgroups of symmetry groups.

- Video/Audio
- Discrete groups of motions; Abstract group actions
- Group actions
- Continued

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

Sylow theorems. Study of permutation groups.

- Video/Audio
- Alternating group structure
- Rings
- Continued

Examples of rings. Basic properties and constructions.

- Video/Audio
- Part 1
- Part 2
- Part 3

Quotient rings. Integral domains, fields of fractions.

- Video/Audio

- Video/Audio
- Part 1
- Part 2
- Part 3

Gauss’ lemma. Eisenstein’s criterion. Algebraic integers.

- Video/Audio
- Part 1
- Part 2
- Part 3

Dedekind domains. Ideal class groups.

- Video/Audio
- Part 1

- Video/Audio
- Part 1
- Part 2

- 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

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.