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.

Permutations

Cosets, Z/nZ.

Quotient groups, first isomorphism theorem

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

Abstract linear operators and how to calculate with them

Properties and construction of operators.

Orthogonal groups

Isometrics of plane figures

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

Group actions

Basic properties and constructions. Groups acting on themselves by left multiplication. Groups acting on themselves by conjugation.

A5 and the symmetries of an icosahedron

Sylow theorems. Study of permutation groups.

Rings

Examples of rings. Basic properties and constructions.

Extensions of rings

Quotient rings. Integral domains, fields of fractions.

Special lecture

Euclidean domains, PIDs, UFDs

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

Structure of ring of integers in a quadratic field

Dedekind domains. Ideal class groups.

Wrap-up

Class Materials