MATH 116 Basic Algebraic Structures (Spring 2006)

# MATH 116: Basic Algebraic Structures

• Announcements:
• The make-up exam is on 24 May 2006, Wednesday at 9.30 in M-102. The contents is everything we covered.
• The final exam is on 22 May 2006, Monday at 13.30. The exam places are the same with the midterm exam places.
• The contents of the final exam is everything covered. See below for the section numbers and skipped topics.
• The contents of the second midterm exam is as follows: 3.4, 4.1, 4.4, 4.5, 5.1, 5.2, 6.2, 7.1, 7.2.
(Notes: Section 4.4: Exclude 4.8, 4.9, 4.17, 4.19 and 4.20.
Section 4.5: Only 4.21 and 4.22.
Section 6.2: Only 6.7 and 6.8 in the context of isomorphisms.)
• The exam places are same as the first exam.
• Recitation times are: 25 April Tuesday 17.40 P-1, or 27 April Thursday 17.40 U-1.
Same problems will be solved, so you can go to the more convenient one.
• IMPORTANT: The date of the second midterm exam is changed to 2 May 2006, Tuesday.
• If you are a student taking MATH 116, then you can subscribe the e-mail group.
• Lectures:
• Section 01: (SF) Monday 11:40-12:30 (M-102) Wednesday 13:40-15:30 (M-102)
• Section 02: (T�) Monday 8:40-10:30 (M-104) Wednesday 12:40-13:30 (M-104)
• Section 03: (AB) Wednesday 10:40-12:30 (M-102) Friday 12:40-13:30 (M-102)
• 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:

Section 01: Sergey Finashin (M-137)
Section 02: Turgut �nder (M-140)
Section 03: Ay�e Berkman (M-242)
All Sections: �zcan Kasal (M-232), P�nar U�urlu (M-232)

### Exams:

Exam 1: 21 March 2006, Tuesday 17:40
Contents: 1.4, 1.5, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 3.1, 3.2, 3.3 (Theorem 3.22 and the rest of section 3.3 are not included).
Exam 2: 2 May 2006, Tuesday 17:40
Contents: 3.4, 4.1, 4.4, 4.5, 5.1, 5.2, 6.2, 7.1, 7.2. (See above or below for the excluded parts.)
Final: 22 May 2006, Monday (Contents: Everything)

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

### 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 + Exam 1 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 + Exam 2 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: 1-27

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.

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Last updated on 11 May 2006.