**MATH 520, Fall 2007**,
12:00-12:50, M W F, DU 328

**Professor John Beachy**,
Watson 355, 753-6753

**Office Hours**:
9:00-10:00 MWF in Watson 355,
1:00-1:50 W in DU 464,
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 600 points:
200 points for homework,
200 points for a two hour exams, and
200 points for the final exam.

**FINAL EXAMINATION:**
The final exam is scheduled for
Monday, December 10, 12:00-1:50 pm

**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**

2007
Monday Wednesday Friday M Tu W Th F
1.1 1.2 1.3 AUG 27 28 29 30 31
Holiday 1.3 1.4 SEP 03 04 05 06 07
2.1 2.1 2.2 10 11 12 13 14
2.2 2.3 2.3 17 18 19 20 21
2.4 2.4 EXAM I 24 25 26 27 28
2.5 2.5 2.6 OCT 01 02 03 04 05
2.6 2.7 2.7 08 09 10 11 12
2.9 2.9 EXAM II 15 16 17 18 19
3.1 3.2 3.3 22 23 24 25 26
3.4 3.4 3.5 29 30 31 01 02
3.5 4.1 4.1 NOV 05 06 07 08 09
4.2 4.2 4.2 12 13 14 15 16
4.3 Holiday Holiday 19 20 21 22 23
4.3 4.3 4.4 26 27 28 29 30
4.4 4.5 4.5 DEC 03 04 05 06 07
FINAL 12-1:50 PM 10 11 12 13 14

**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.
One assignment late in the semester will be given under the rules
for a takehome exam--no consultation with other students.