Math 101 - Topics in Algebra- Fall 2010
Dartmouth College
Week | Brief Description |
---|---|
1 | Review of groups, rings and fields. Substructures: subgroups, subrings and subfields. Structure preserving maps: homomorphisms and isomorphisms. |
2 | Group homomorphisms, kernel, cosets. Lagrange theorem, normal subgroup. Isomorphism theorems, composition series, Holder's program. Ring homomorphisms, ideals, kernels factor rings. Direct products. |
3 | Symmetric groups, conjugacy classes, groups acting on sets and Polya's enumeration theorem. |
4 | Class equation and the Sylow theorems. |
5 | Vector spaces, subspaces, linear transformations, bases, direct sums. Inner product spaces, normed spaces, orthogonal sets. |
6 | Gram-Schmidt process. Matrices, row-operations, rank. Matrices associated to linear transformations. |
7 | Change of basis, determinants. Polynomials, unique factorization |
8 | Cayley-Hamilton Theorem. Normal forms, characteristic equation, Smith normal form, Jordan canonical form, rational canonical form |
9 | Introduction to modules |