#
Math 551
Abstract Algebra

## Fall 2010

Prof. C. Weibel
Meets TTh 3; 12:00-1:20 in H425
Main Text: N. Jacobson, *Basic Algebra I, II*
Homework Assignments (Fall 2010)

This is a standard course for beginning graduate students. It covers:

Group Theory | 10 lectures | Sept.1-Oct.4 |
Midterm |

Basic Ring & Module Theory | 9 lectures | Oct.7-Nov.1 |

Modules over a PID | 3 lectures | Nov.4-11 |

Bilinear Forms | 3 lectures | Nov. 15-18,29 |

Artin-Wedderburn & Maschke theorems | 4 lectures | Dec.2-13
| Final Exam |

**Group Theory:** Basic concepts, isomorphism theorems,
normal subgroups, Sylow theorems, direct products and free products of groups.
Groups acting on sets: orbits, cosets, stabilizers.
Alternating/Symmetric groups.

**Basic Ring Theory:** Fields, Principal Ideal Domains (PIDs),
matrix rings, division algebras, field of fractions.

**Modules over a PID:** Fundamental Theorem for abelian groups,
application to linear algebra: rational and Jordan canonical form.

**Bilinear Forms:** Alternating and symmetric forms, determinants.
Spectral theorem for normal matrices,
classification over **R** and **C**.
(
Class supplement provided)

**Modules:** Artinian and Noetherian modules.
Krull-Schmidt Theorem for modules of finite length.
Simple modules and Schur's Lemma, semisimple modules.
(from *Basic Algebra II*)

**Finite-dimensional algebras:** Simple and semisimple
algebras, Artin-Wedderburn Theorem, group rings, Maschke's Theorem.
(
Class supplement provided)

*Prerequisites:* Any standard course in abstract algebra for
undergraduate students.

Last updated: November 1, 2010