Math 412 Section 2: Reading, practice problems and assignments for Section 2

Text: Hungerford, "Abstract Algebra: an introduction," second edition. Below are the individual reading and practice problems, due on the date listed. These problems are not to be turned in. The graded group assignments, which are the same for sections 1 and 2 and are due each friday starting Sept 18, can be found here, on the general Math 412 website.

SUPPLEMENTAL ENRICHMENT PROBLEMS FOR STUDENTS WHO WANT TO UNDERSTAND MORE ABSTRACT PRINCIPALS UNDERLYING THE COURSE:

Read 6.1 Do problems 6.1 #2, 7a, 7c, 13, 15

Read 6.2 Do problems 6.2 #1, 3, 4, 5, 6

Read 6.3 Do problems 6.3 #1, 3, 4, 5, 6

For each section, you may read and turn in the problems for that section (or two or all three). I will grade them and we can discuss. Chapter 6 describes a common generalization of the two main examples of rings (and congruence) we've studied so far, so for those of you who enjoy the abstract stuff, reading chapter 6 may actually help solidify your understanding of the previous material. Thus, if you are totally on top of the course (have done all assigned A and B problems, and find them easy(ish)), reading Chapter 6 and doing the exercises above is not a bad way to supplement your studying for the exam.

If anyone would like a similar additional assignment on material related to groups (constructing ring quotients G/K similar to ring quotients R/I from Chap 6), let me know.

FINAL EXAM: Dec 17 10:30-12:30 in 1200 CHEM. Will cover Chap 1,2,3, 4 (except 4.5), 5, 7.1--7.5, and 7.9 (up to section called "alternating group). The exam will emphasize material from Chapter 7 but will cover everything.
REVIEW MATERIALS:

A true-false practice sheet on Chapter 7, and on Chapter 4 and 5

Vocabulary Sheet for Chap 7

Old Exams: Exam 1 , Exam 2 ,

Quizzes: Quiz 1 , Quiz 2 , Quiz 3 , Quiz 4 , Quiz 5 , Quiz 6 , Quiz 7 , Quiz 8, Quiz 9 , Quiz 10, Quiz 11 ,
OFFICE HOURS DURING EXAM WEEK: Monday 12-1 (as usual), Review Q/A Session Tuesday 10-12 in East Hall 3096, and by appointment. Students from either section are welcome. Also: you are encouraged to also attend the other instructor's review session tuesday 6-8 (check his website for location).

REGULAR PRACTICE PROBLEMS EXPECTED FROM ALL STUDENTS:

• For Friday Dec 11: Read 7.9 Practice: 7.9 # 1, 2, 3 (Easy(ish) Challenge: 7.6 # 4, 7, 8, 9)
• For Wednesday Dec 9: Reread 7.5, practice 7.5 #1, 3, 4, 5, 7, 8, catch up on problems from 7.3, 7.4 (Easy(ish) Challenge: Read 7.6)
• For monday Dec 7: Read 7.5, Practice problems: 7.3 #15, 18, 19; 7.4 #6, 13; 7.5 #2; Easy(ish) Challenge: 7.3 #22, 46, 48
• For Friday Dec 4: Reread 7.3/7.4, paying special attention to the proof that every group is isomorphic to a subgroup of a permutation group. The key idea: how does each element of G induce a permutation on the set G? Practice problems: 7.3 #12, 13, 7.4 # 1, 3, 4, 5 PREPARE FOR QUIZ.
• For Wednesday Dec 2: READ 7.4 Practice problems 7.1 #13, 7.2 #8, #9b, #18 7.3 #5, #6, #7
  (NOTE: THE GROUP HOMEWORK CONTAINS TWO PROBLEMS FROM 7.5. These use only Lagrange's Theorem: The order of subgroup divides the order of a group.)
• For Monday Nov 30: Practice problems: 7.1 #8, 9, 11; 7.2 #12, 13; 7.3 #1, #3. Also, finish the Worksheet we worked on Wednesday in class.
• For Wednesday, Nov 25: Read 7.3. Practice problems 7.2: # 1, 2, 3, 4, 7.
• For Monday, Nov 23: Reread 7.1, read 7.2. Practice problems: 7.1 # 2, 3, 4.
• For Friday: Read 7.1 carefully! Do exercise 7.1 #1. A quiz on 7.1 is likely.
• For Monday Nov 16: Study for Tuesday's exam! Monday will basically be an office hour, in which I will answer your questions that came up during studying, including the last two handouts: True-False Practice Sheet and Wednesday's Practice problems already assigned. You can also ask your friends in the other section for their true-false quiz (which I distributed in class Friday, stapled to the back of my own, but I do not have an electronic version). The format of the exam will be similar to the first exam, so you might want to look it over (though it may be harder!). The best way to study is practice and make sure you have a large arsenal of examples and non-examples of all the concepts discussed. After the review sheets, re-do the group homeworks, make sure you can do all the assigned A-problems from chapters 4 and 5, then all the assigned B-problems, then additional A-problems, then additional B-problems.
• By Friday the thirteenth: Practice problems (same as what was sent by email) for discussion after the QUIZ! One quiz problem is direct from group homework. ALSO: Please be sure you came to a full understanding regarding the problems you began in class on Monday and Wednesday, which are typical of something you might expect to see on Tuesdays exam.
  Monday's problem: Among all four rings of the form F[x]/(p(x)), where F is a field of two elements and p(x) is degree two, which (if any) are isomorphic to eachother.
  Wednesday's problem: Find a condition on the polynomial p(x) which is satisfied if and only if F[x]/(p(x)) has no non-zero nilpotent elements. (Recall: a nilpotent is an element r such that r^n = 0 for some n.)
• By Wednesday Nov 11: Practice problems 5.3 #3, 4, 5, QUIZ ON CHAPTER 5.
• By Monday November 9: Read 5.3. Practice problems 5.1 #10, 5.2 #7, 10 5.3: #1,2 (if you haven't done the previous exercises, do them first---be sure you can do all assigned problems from section A before stressing about the B problems.
• By Friday Nov 6: Reread 5.1 and 5.2. Also read 5.3. Practice problems 5.1 #5, 6, 7, 12 5.2: #1, 5
• By Wednesday Nov 4: Read 5.1, 5.2 (carefully...it's hard!). Practice problems 5.1: #1,2,3, 4.
• By Monday Nov 2: Read 4.4, 4.6. Practice 4.4 #2c, #3d, 6, 8 4.6 #1a, 2 EXTRA CREDIT: Read 4.5 #5, 7 (you can turn in written solutions or show me on the blackboard).
• By Friday: 4.2 (some from problems 5 and 6---make sure you understand how to find gcd's of polynomials with exotic coefficients and how to work the euclidean algorithm backwards to write it as a linear combination), 8, 4.3: 4, 7, 9, 10. Challenge: 4.3 # 21
• By Wednesday Oct 28: Reread 4.2, read 4.3. Practice problems: 4.2 #3, 4, 5a, 5f; 4.3 #1, 2, 3b
• By Monday Oct 26: Read 4.1, 4.2 (QUIZ). Practice problems: 4.1: 1b, 3a, 4, 5a, 6
• For Friday Oct 16: Keep reading and studying, doing the practice problems already assigned. The last quiz was a *serious wake-up call* for your professor and I hope to the 50% of you who scored 50% or less! No new problems assigned until after the exam, so you can all get caught up. The exam next week will not be any easier than planned but I will have extra office hours and encourage you all to get serious about the reading and practice problems so we can discuss FRIDAY IN CLASS. If you are on top of the course, and want some challenge problems, try: p 65 #25, 26
• By Wednesday Oct 14: Catch up on all practice problems. Try also: 3.1 # 24, 25; 3.2 #31, 3.3 # 7.
• By monday Oct 12: Reread 3.3. Practice problems: 3.2, #12, 3.3 #3, 6, 8, 10,
• By Friday Oct 9: Read 3.3. Practice problems: 3.1 #4, 6, 9, 12, 15; 3.2: #2, 7, 8, 10, 11, 13; 3.3 #2 (Many of these you did in class wednesday)
• By Wednesday Oct 7: Read 3.1 and 3.2 carefully (quiz!!). Practice problems 3.1 # 1, 3, 5 and 3.2 # 1,3, 5
• By Monday Oct 5: Get caught up on all previous practice problems from chapter 2 and 12. Be sure you understand the principle behind Public key encryption. Also: Practice problem page 40 #6, and any others you like.
  First EXAM is Wednesday October 21, at 7-8:30, covering chapters 1,2,3 and 12.
  The fourth GROUP HOMEWORK due date is moved to Wednesday October 14.
• By Friday Oct 2: Read Chapter 12, and (lightly) the wikipedia entry on the "RSA algorithm" Practice p405 # 3a. pp 39 # 3, 5
  Challenge problem: page 40 #13
• By Wednesday Sept 30: Study for quiz on solving harder problems in modular arithmetic, including finding multiplicative inverses. For practice, pp 39-40: #2, #7.
• By Monday Sept 28: Practice Problems pp 29-30 2bcd, 8ab, 10, and pp 39 #1.
• By Friday Sept 25: Practice Problems pp 29-30 #11, #20, pp 35-36 # 3, #5, pp 39 #4.
• By Wednesday Sept 23: Read 2.3
  Practice problems: page 35-36: #1a, 2a, 6, 7, 10ab
  Challenge: p 36 #11
• By Monday Sept 21: Read 2.2
  Practice Problems: page 29 #1, 2, 3, 19
  Challenge Problem: page 29 #21
• By Friday Sept 18: GROUP ASSIGNMENT DUE (See assignment on general Math 412 site)
  Read Section 2.1
  ALSO: QUIZ to make sure everyone mastered all the group homework.
  Practice problems: page 18 #1, 6, 9, 13, 14
  Challenge problem: page 19: #24.
• By wednesday Sept 16: Read Appendix C
  Practice/quiz problems: p 525 #1, 3,
• By monday Sept 14: 1.2, 1.3, 1.4, Appendix C
  Practice/quiz problems: page 12-14, #1 (a,c, e, g), #3, # 5
• By friday Sept 11: Read Appendix A, B, Sections 1.1, 1.2
  Practice/quiz problems: page 6, #1, # 5