Math 101 - Topics in Algebra- Fall 2010

Dartmouth College

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The following is a tentative syllabus for the course. This page will not be updated regularly during the term. For un uptodate description of the material discussed in clas see the homework page

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

Last Updated Sept. 16, 2010 by R. Orellana