MATH 520, Fall 2004, 11:00-11:50, M W F, DU 254

Professor John Beachy, Watson 355, 753-6753
Office Hours: 10:00-10:50 MW in Watson 355 and 12:00-12:50 in DU 254, or by appointment

### SYLLABUS

COURSE: ALGEBRAIC STRUCTURES I (3) Group theory including the Sylow theorems, the basis theorem for finite Abelian groups. Polynomial rings, field theory, Galois theory, solvable groups, and solvability of equations by radicals.

PREREQUISITE: MATH 421 or consent of department.

TEXT: Lecture Notes (available in class); the textbook Algebra, by Hungerford, is recommended as a reference.

GRADING: Semester grades will be based on 500 points: 200 points for homework, 100 points for a midterm exam, and 200 points for the final exam.

FINAL EXAMINATION: The final exam is scheduled for Wednesday, December 8, 10:00-11:50 a.m.

SYLLABUS:
CHAPTER 1: 1.1 Groups; 1.2 Subgroups; 1.3 Examples; 1.4 Homomorphisms and factor groups
CHAPTER 2: 2.1 Automorphisms, semidirect products; 2.2 Conjugacy; 2.3 Group actions; 2.4 The Sylow theorems;
2.5 Finite Abelian groups; 2.6 Solvable groups; 2.7 Simple groups; 2.9 Groups of small order
CHAPTER 3: 3.1 Fields, polynomials; 3.2 Irreducible polynomials; 3.3 Algebraic extensions; 3.4 Splitting fields;
3.5 Structure of finite fields
CHAPTER 4: 4.1 The Galois group of a polynomial 4.2 Multiplicity of roots; 4.3 The fundamental theorem of Galois theory;
4.4 Solvability by radicals; 4.5 Cyclotomic extensions

TENTATIVE SCHEDULE OF LECTURES

```Monday      Wednesday    Friday           M Tu  W Th  F
1.1           1.2         1.3       AUG 23 24 25 26 27
1.3           1.4         1.4       SEP 30 31 01 02 03
Holiday        2.1         2.1           06 07 08 09 10
2.2           2.2         2.3           13 14 15 16 17
2.3           2.4         2.4           20 21 22 23 24
2.5           2.5         2.6           27 28 29 30 01
2.6           2.7         2.7       OCT 04 05 06 07 08
2.9           2.9       MIDTERM         11 12 13 14 15
3.1           3.2         3.3           18 19 20 21 22
3.4           3.4         3.5           25 26 27 28 29
3.5           4.1         4.1       NOV 01 02 03 04 05
4.2           4.2         4.2           08 09 10 11 12
4.3           4.3         4.3           15 16 17 18 19
4.4          Holiday     Holiday        22 23 24 25 63
4.4           4.5         4.5       DEC 29 30 01 02 03
FINAL  10-11:50 AM         06 07 08 09 10
```
HOMEWORK I encourage you to study in groups, and you may discuss homework problems with other students. You should write up your own solutions--direct copying is unacceptable. As a rough guideline for writing up solutions to homework problems, you should include enough detail so that (i) you can convince me that you understand the solution and (ii) you can understand your solution when you study for the final exam. Two assignments will be given under the rules for a takehome exam--no consultation with other students.