MATH 116 Basic Algebraic Structures (Spring 2007)

# MATH 116: Basic Algebraic Structures

• (5 June) You can see your second exam papers on 8 June Friday between 1.00 and 3.00 in M-201.
• (4 June) The make-up exam is on 11 June Monday at 9.00 in M-102. The exam will cover all the course topics.
• (26 April) First midterm results are announced on the announcement board number 6 on the first floor.
• Lectures:
• Section 01: (GE) Monday 11:40-12:30 (M-103) Wednesday 13:40-15:30 (M-103)
• Section 02: (AB) Monday 8:40-10:30 (M-102) Wednesday 12:40-13:30 (M-102)
• Section 03: (KZ) Wednesday 10:40-12:30 (M-103) Friday 12:40-13:30 (M-103)
• Textbook. Elements of Modern Algebra, Jimmie Gilbert and Linda Gilbert, Brooks/Cole, fifth edition, 2000. (Five copies are available in the reserve section of the library.)

### People and Office Hours:

Section 01: G�lin Ercan (M-139) Thurs 15.40-17.00
Section 02: Ay�e Berkman (M-242) Mon 10.40-11.30
Section 03: Konstantin Zheltukhin (M-137) Mon 9:40-11:30 and Wed 14:30-16:30
All Sections: M. Onur Fen (M-201) Thurs 10.40-12.30
Hakan G�rba�lar (M-201) Wed 15.00-16.00 and Fri 12.00-13.00

### Exams:

Exam 1: 3 April 2007, Tuesday 17:40 Contents: Everything we covered between 1.4 and 3.3 (inclusive).
Exam 2:15 May 2007, Tuesday 17:40 Contents: TBA
Final: 3 June 2007, Sunday, 13.30 (Contents: Everything)

Exam 1 - 30 points
Exam 2 - 30 points
Final - 40 points

### TENTATIVE Outline of the course:

 Weeks Topics Sections 1 Binary operations 1.4, 1.5 2 Integers, divisibility, prime factors, greatest common divisor 2.1-2.4 3 Congruence of integers, congruence classes 2.5, 2.6 4 Groups, subgroups 3.1, 3.2 5 Subgroups, cyclic groups 3.2, 3.3 6 Cyclic groups, isomorphisms 3.3, 3.4 7 Permutation groups 4.1. 8 Normal subgroups, quotient groups 4.4*, 4.5** 9 Rings 5.1. 10 Ring isomorphisms, integral domains 5.2, 6.2*** 11 Fields, the fields of real and complex numbers 5.2, 7.1, 7.2 12 Polynomials, divisibility 8.1, 8.2 13 GCD, factorization 8.2, 8.3 14 Zeros of a polynomial 8.4****

(*) (4.8), 4.9, 4.17, 4.19 and 4.20 will be skipped.
(**) Only 4.21 and 4.22.
(***) We will cover 6.7 and 6.8 (only in the context of isomorphisms).
(****) 8.30 - 8.33 will not be covered.

### Exercises:

Solving exercises (NOT reading the solutions but really solving them) is an essential part of every math course and MATH 116 is no exception. Therefore to start studying for the exam the day before is not a good idea, because there is simply no time to work on all the exercises. We recommend you to solve the exercises of a section within the same week that the section is covered in the class. Good luck and have fun!

Suggested Problems:

Section 1.4: 1, 2a, 2f, 3, 4, 5, 6, 8, 9, 10e, 10k, 13

Section 1.5: 2a, 2b, 3d, 3e, 3i, 20, 22, 23, 24, 26

Section 2.1: This seciton is optional.

Section 2.3: 16, 17, 18, 20, 21, 22, 23, 24, 25, 27, 29

Section 2.4: 2a, 3c, 3j, 4a, 5, 7, 9, 10, 11, 13, 14, 15, 16, 19, 27

Section 2.5: 5, 6,16, 19, 20, 21, 22, 23a, 25, 26, , 27, 28

Section 2.6: 5a, 6e, 8c, 8f, 9, 10c, 16, 18, 22

Section 3.1: 1, 2, 4,5, 6, 7, 11, 15, 17, 23, 28, 31, 33, 45

Section 3.2: 2b, 2c, 3, 5a,6a, 10, 11, 13a, 14b, 17, 23, 24, 28, 35

Section 3.3: 7, 8, 9 a, 9b, 10a, 10b, 12, 14, 15, 18, 23, 24

Section 3.4: 1, 3, 6, 10, 11, 16, 17, 18, 20(a, b), 26, 28.

Section 4.1: 1(a, e, h), 2(a, b, f), 3, 4, 5, 6, 7, 8, 13, 14.

Section 4.4: 5, 6, 7, 8, 25, 26, 27, 30, 32, 35.

Section 4.5: 2, 4, 10, 15, 16.

Section 5.1: 1(a, b), 2(b, c), 5, 7(a, b, e, f), 8(c, d), 9, 10, 12, 16, 19, 21, 22, 24.

Section 5.2: 1(a, b), 2, 3, 5, 6, 8, 9, 10, 11, 12, 15(a, c), 18, 19.

Section 6.2: 1, 2, 3, 7, 8, 10, 12, 15 (Replace "homomorphism" with "isomorphism".), 19.

Section 7.1: 15, 16, 17, 18, 19, 20.

Section 7.2: 1, 4, 7, 11, 12, 13b, 14d, 16, 17, 18, 19, 20.

Section 8.1: 3e, 5, 7, 8, 9, 15 (replace epimorphism with isomorphism).

Section 8.2: 3, 6, 10, 14, 15, 16, 21.

Section 8.3: 1, 2, 3a, 5, 6, 7, 11 (Assume problem 9 to solve 11.).

Section 8.4: 1 (c,e), 2 (a,c), 4, 7, 15, 17a, 18, 19, 20, 21, 23, 24.