Introduction to Modern Algebra (Math412, section 002)


Welcome to Introduction to Modern Algebra! Here is the syllabus. Let's emphasise that the Monday homework is due at the _beginning_ of the class!

Office hours: M 10-11, Th 1-2, F 10-11 in East Hall, 3855.

The first midterm: Monday, February 11, in class.
The second midterm: Wednesday, April 2, in class.
The final exam: Wednesday, April 23, 4-6pm. Please, double check with me! See Winter 2008 Final Examination Schedule .
In the first class (Friday, Jan 4) We covered section 1.1 and parts of section 1.2. Homework: ex. 5,6 page 6 and ex. 1,3,5,7,9,11 page 12. Note: the deadline of the first homework is extended to Friday, Jan 11 since not everyone in class got the textbook.
Monday, Jan 7: We started with discussion about ex. 4, page 6. After that we continued on 1.2 (definition of gcd, Thm. 1.3 and Cor. 1.4). For Wednesday prepare exercises 16, 17, 20 and 35. "Prepare" means you should know how to do it, but you do not need to write down the solutions. Homework for Monday (this one you should write down!): 21,31,32,34. Note that I might (and most likely will) add more exercises on Wednesday!
Wednesday, Jan 9. We finished section 1.2 and (unfortunatelly very briefly) covered Appendix C (Mathematical Induction). Problems to think about by Friday (again, no need to write them down): 2,3,4,5,6,7,8,9,10,11,12 on page 525, and (again) ex. 17, 35 on page 14.
Friday, Jan 11: section 1.3 (Primes and factorization). We almost finished the section, the only thing to do is to prove Thm 1.11. Problems to think about: 20, 23 (and 4 if you feel like it).
Monday, Jan 14: section 2.1 (Congruence). See an interesting article from Wikipedia about prime numbers . Problems to think about by the next class: proof of Thm 2.2 (2), ex. 11, 12. Also, think what the is congruence class of 0 modulo 1?
Wednesday, Jan 16. section 2.2 (Modular arithmetic). Problems to think about: proof of Thm. 2.6 and Thm.2.7.
Friday, Jan 18. We finished section 2.2 and we started section 2.3 (structure of Z_p when p is prime). Do problems in section A, page 35-36 together with ex.8 a),b) , 9b)c), and 10. Expext a quiz next Wednesday.
Monday, Jan 21. No class. MLK day.
Wednesday, Jan 23: section 2.3. Problems to think about:(1)prove that in a ring 0 *a=0 (2) section 3.1 exercises 1,4,5,6.
Friday, Jan 25: section 3.1 (Definition and examples of rings). Interesting exercises 1-15. More interesting problems: 20, 22, 24. Even more interesting: 37,36,35.
Monday, Jan 28: section 3.2 (Properties of Rings). Prepare proofs of three theorems from section 3.2!
Wednesday, Jan 30: section 3.3 (Isomorphisms and Homomorphisms).
Friday, Feb 1: section 4.1(Polynomial Arithmetic) Note that there is an extra credit problem (due next Wednesday): show that distributive law holds in polynomial ring and that multiplication and addition are associative!
Monday, Feb 4: section 4.2 (Divisibility in F[X])
Wednesday, Feb 6: section 4.3 (Irreducibles and Unique Factorization)
Friday, Feb 8: Review. Samle test . Solutions.
Monday, Feb 11: Midterm Exam. Solutions.
Wednesday, Feb 13: section 4.3 (Irreducibles and unique factorization). Extra credit problem (due Monday): ex. 24 page 99.
Friday, Feb 15: section 4.4 (Roots and reducibility).
Monday, Feb 18: section 4.6 (Irreducibility in C[x] and R[x])
Wednesday, Feb 20: section 5.1. (Congruence in F[x])
Friday, Feb 22: We finished section 5.1. We did some examples similar to exercises 2,3,4,5,6 and 7 on page 123. We also proved Thm. 5.6, section 5.2 (Congruence class arithmetic).
Monday, Feb 25 - Friday, Feb 29: No class (Spring break!)
Monday, March 3: section 5.2 (Congruence class arithmetic). By Wednesday prepare one of the exercises from part A page 128.
Wednesday, March 5; section 5.3 (Structure of F[x]/(p(x)) when p(x) is irreducible). On Friday we will start talking about ideals, so please look at the exercises 1-6 in 6.1.
Friday, March 7 and Monday, March 10: section 6.1 (Ideals)
Wednesday, March 12 and Friday, March 14: section 6.2 (Quotient rings)
Monday, March 17: Groups (section 7.1).
Wednesday, March 19 and Friday, March 21: Properties of groups (section 7.2) . Prepare proof of Thm. 7.8 and ex. 17, 26 and 28 from section 7.1.
Monday, March 24 and Wednesday, March 26 and Friday, March 28: section 7.3 (Subgroups). For Friday, prepare the following problems: (1) What is the center of the group Gl(2,R)? (invertible 2 by 2 matrices with real entries) (2) Is (Q, +) cyclic? (3) proof of Thm. 7.15.
Monday, March 31: Review. Some practice problems about ring homomorphisms and isomorphisms. Here are the solutions . Sample exam and solutions.
Wednesday, April 2: Exam 2. Sections 4.3- 7.3. (without 4.5 and 6.3) together with some problems concerning ring isomorphisms and homomorphisms. Exam 2. Solutions Friday, April 4: section 7.4 (group homomorphisms and isomorphisms) and the beginning of section 7.5 (Congruence). For Monday prepare proofs of Thm. 7.19, 7.22, 7.23 and 7.24.
Monday, April 7: section 7.5 (Lagrange Thm).
Wednesday, April 9: section 7.6 (Normal subgroups). For Friday prepare proof of Thm. 7.33.
Friday, April 11: section 7.7 (Quotient groups) and parts of section 7.8 (Quotient groups and homomorphisms). We finished with The First Isomorphism Theorem (Thm 7.42)
Monday, April 14: Review. We will work on this sample test . Here are other practice problems

Homework Assignments

Number Due date Problems Solutions Median
1 Friday, January 11 section 1.1 ex. 5,6 section 1.2 ex. 1,3,5,7,9,11. solutions 1 9
2 Monday, January 14 section 1.2 ex. 21,22,25,26,27,31,32,34 solutions 2 9
3 Wednesday, January 23 section 1.3 ex. 5,12,17,19,20, section 1.4 ex. 7,10,12 solutions 3 8
4 Monday, January 28 section 2.1 ex. 28,29,30, section 2.2 ex. 11, section 2.3 ex. 7,8,9,10 solutions 4 8
5 Monday, February 4 sec. 3.1 ex. 21, 25,26,35, sec.3.2. ex.5,10,31, sec. 3.3 ex.1 solutions 5 8.75
6 Monday, February 11 sec.3.3 ex. 10,25,28,33 sec.4.1 ex. 1,6,18 sec.4.2 ex. 5 c) d) e) solutions 6 9
7 Monday, February 18 sec.4.2 ex. 10,12,13 sec. 4.3 ex. 9,13,14 Extra credit: sec. 4.3 ex. 24 solutions 7 10
8 Wednesday, March 5 sec.4.4 ex. 8,10,13,16,17 sec. 4.6 ex. 6,7,8 solutions 8 9
9 Monday, March 10 sec.5.1 ex. 8, sec. 5.2 ex. 6, 13, 14, sec. 5.3 ex. 2,3,10,11. solutions 9 9.25
10 Wednesday, March 19 sec.6.1 ex. 13b, 14, 15b, 16, 43, sec. 6.2 ex. 7,14,31 solutions 10 8
11 Wednesday, March 26 sec.7.1 ex. 4,13, 21, 27 sec. 7.2 ex. 7, 12, 18, 27 solutions 11 9
12 Monday, March 31 sec.7.3 ex. 9,10,12,15,16,20,21. solutions 12 9
13 Wednesday, April 9 sec.7.4 ex. 1,2,5,6,10,13,28 solutions 13 10
14 Monday, April 14 sec.7.5 ex.2,3,12 sec 7.6 ex. 3,6 sec. 7.7 ex. 1,4,15 solutions 14 10

Quizzes

Number Date Material Median
1 Monday, January 28 section 2.2 10

Have a great term!