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.
- 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
- 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.
- More on quotients; Vectorspaces
Abstract linear operators and how to calculate with them
Properties and construction of operators.
- Bases and vectorspaces; Matrices and linear transfs
- Bases; Matrices
- Eigenvalues and eigenvectors
- Review for midterm; Orthogonal group
- 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.
- Discrete groups of motions; Abstract group actions
- 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.
- Alternating group structure
Examples of rings. Basic properties and constructions.
Extensions of rings
Quotient rings. Integral domains, fields of fractions.
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.
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.