COURSE  SYLLABUS -Math 330 Spring 2001 

ABSTRACT ALGEBRA 

Instructor:  B Gray
Office 508 SEO   Phone 6-4828 brayton@uic.edu
Office Hours 10:00 MWF and by appointment
 
Course Meeting: 11 AM MWF, Taft Hall 212

GRADING:
EXAM 1- 25% Chapters 0-7 (approximately)
EXAM 2- 25% Chapters 8-15 (approximately)
FINAL -50% Comprehensive

Homework assigned will be collected on Monday unless otherwise notified.
A good homework record improves a border line grade.

TEXT: Text: J. Gallian, Contemporary Abstract Algebra, 4th edition.
PREREQUISITE: A grade C or better in Math 320.
Course covers Chapters 0-23 in text.

The aim of the course is to introduce students to the basic abstract algebraic 
structures such as groups, rings and fields. Students will be expected to
read the text and comprehend proofs, as well as to do problems involving
both computations and proofs.

Note: calculators will not be used in this course.

LIST OF TOPICS:
A brief review of sets, mappings and equivalence relations. Properties of 
integers.
Groups: Subgroups, homomorphisms, cyclic groups, permutations, matrix groups, 
cosets, Langrange's Theorem, normal subgroups and factor groups.
Rings: Integral domains, homomorphisms, ideals, factor rings, polynomial rings,
factorization of polynomials.
Fields: Finite extensions and their connection with polynomial rings, 
construction of finite extensions, application to geometric
constructibility problems.

READING ASSIGNMENTS:
Integers and Equivalence Relations, Chapter 0 Groups, Chapters 1-10
Chapters 1-4, all pages Chapter 5, p.90-102; Ch 6, p.115-122; Ch 7, p.132-137; 
Chapter 8, p.149-156 and Example 7; Ch 9, p.171-178; Ch 10, p.192-203
Chapter 12, p.225-230; Ch 13, p.236-242; Ch 14, p.248-255; Ch 15,
p.266-272; Ch 16, p.279-285; Ch 17,p.291-300 Chapter 20,
p.341-346; Ch 21, p.359-364; Applications:Chapter 23, p.383-387.

HOMEWORK: unless otherwise notified, you will be expected to do the following 
homework problems:

Chapter 0: Practice: p.21-24, #1,3,7,18, 19,20,21,25,35,43. 
Hand-in: p.22, # 10; p.24, # 21, 44. 

Chapter 1:Practice: p.35-37, #1, 3, 5, 7, 9, 17, 21, 23; p.51-54, #1, 3, 5, 11,
13, 15, 19, 21; Hand-in: p.35-37, #10, 22; p.51-54, #8, 16, 24, 28, 34 

Chapters 3 and 4. Practice: p. 65-69, #13,5,7,11,13,21,23,27,29,43,47,49,51. 
p. 79-81: #1,3,5,9,17,19,27,37,43,61. Hand-in: p.65-69, #8,28,30, 38(b), 44,48 
p. 79-81, #34,36,44,60 

Chapters 5 and 6: Read Ch 5, p. 90-102; Ch 6, p.115-123. Practice: p.107-110: 
#1,3,5,9,11,13,15, 17, 21,25,27,37,41,53 p. 126-128: #3,5,7,17,23,27 Hand-in:
p. 107-110: #18,28,34,36, p. 126-128: #4, 16,24.

Chapters 7 and 8: Read Ch 7, p. 132-137 and p. 140, Th 7.3; Ch 8, p. 149-156 
and Example 7 Practice: p. 142-145: #1,3,5,7,9,11,13,15,17,23,25,27 p.
161-163: #1,3,9,11,13,15,17,25; Hand-in : p. 142-145: #6, 8, 14, 20, 26,
30; p. 161- 163: #8, 20, 26, 30; Review: p.53,#19. Add: Show this group is
isomorphic to U(8). p. 67, #32; p.80, #23 (these two go together) p. 67-68,
#34, 42, 50 p. 79-81, #8, 12, 31, 46, 50 p. 86-88, #8, 16, 26 (See also the 
pictures on p.463.) Review: p. 108, #6 [Hint: Look at the possible cycle
types of elements of 56 or A6] p. 109, #32, 38. Add: In D4 find elements x
and y such that x, y, xy have orders 2, 2, 4. p. 128, #36; p. 142, #12,
18, 22, 24 [For #12, to give a "geometric description" draw a picture in the 
plane] p.161-162, #28, 38 p.168-169, #23,25,26,27,28,30,40,50 [Hint for
p.169, #26: Count the number of elements of order 2.]

Chapter 9: Read p. 171-178 and Theorem 9.5 Practice: p. 185-187: #1,3,5,7,9,
11,17,21,23,25,27 Hand-in : p. 185-187: #4, 12, 16, 20, 26;

Chapter 10: Read p. 192-203; omit p. 201, Examples 15 and 16 Practice: 
p. 203-206, #1, 3, 5, 7, 9, 13, 17, 19, 21, 23, 25, 27, 29 Hand-in : p.
203-206, #2, 6, 8, 18, 26, 44;

Chapters 12, 13: Read p.225-230, p.236-242 Practice: p.230-233, #1, 3, 13,
17, 33, 39 p.242-246, #5 9, 11, 13, 29, 49, 51 Hand-in: p.230-233, #4, 18,
34, 38, 40 p.242-246, #4, 12, 20, 24, 54 Hints: p.231, #4: Use matrices.
p.243, #12: Use the ring Z/n for some n. p.243, #54: Look at the
multiplicative group of the field. p.243, #20. Do not check every element 
of the ring. Take an element (a,b) and see what it means for it to be
idempotent, nilpotent, etc.;

Chapters 14,15: Read p.248-255, p.266-272 Practice: p.255-258, #1,3,5,7,15,
17,23,45; p.274-277, #1, 3, 5, 7, 9, 11, 23, 25, 27, 55 Hand-in:
p.255-258, #4, 18, 48, 50; p. 274-277, #6, 8, 14, 24 Hints: p.255-258,
#18,48, 50: See Examples 10, 11 p.274-277, #14: Read p.268, Ex 5 and use
Exercise #12; Recall group homomorphisms from Z/n to Z/m (e.g. Ch 10,
Example 10); #24. Use p.268, Example 8 and p.275, #22.; Chapters.16, 17:
Read p.279-285, p.291-300. Practice: p.287-289, #1, 3, 11, 13, 15, 19, 23,
27, 29, 31, 33, 35 p.303-305, #7,9,11,13,15,17,19,21,23 Hand-in:
p.287-289, #12,18,32,34,39; p.303-305, #8,10,18 Chapters 20, 21, 23: Read
p.341-346, p.359-364, p.383-387 Practice: p.354-356, #1, 3, 5, 7, 9 p.388,
#1, 3, 5,7,9, 13,21 Hand-in: p.354-356, #4,8 p.367-369, #8,16 p.390, #6