Math 222 -- Course Info

Last modified: Wed May 2 22:01:35 CEST 2001

Questions email Prof. Peter Cholak (Section 1) or Prof. Juan Migliore (Section 2)

This web page will be updated during the course of the semester

• Course Information Handout; Section 1, Section 2
• Errors in the Textbook

Exam Schedule

• Exam 1 - Wednesday, February 28, in class: Sample Exam, Some Sample Exam Solutions, Exam solutions.
• Exam 2 - Wednesday, April 25, in class: Sample Exam, Exam solutions.
• Final Exam - Monday, May 7, 1:45 -- 3:45, in FITZ 356: Sample Final, A Sample Final Solution,.

Course Handouts

• Constructing a pentagon.
• RSA; related Mathematica notebook, solutions (download the file before opening).
• Brief introduction of rings and fields; solutions.
• Irreducible cubics; solutions.
• Examples of groups.

Related Materials (not required)

• An example of the use of the group S_54 in the "real" world.
• The classification of finite groups.
  

Take-Home Quizzes

• Quiz 1, Solutions.
• Quiz 2, Solutions.
• Quiz 3, Solutions.
• Quiz 4 Solutions.
• Quiz 5, Solutions.
• Quiz 6, Solutions.

Course Assignments and Schedule

Warning: The future is subject to change.
The dates the quizzes are given might change.
The syllabus will be updated during the cousre of the semester.

• WEEK 1 - January 17 - 19
  
  Wed. - 1.1: Introduction, 1.1, Start Appendix F: Mathematical Induction
  HW -- 1.1: 4-7, 14,15,17 (16 is a bonus problem)
  
  Fri. - Appendix F: Mathematical Induction
  HW -- Appendix F: 2,4,6,7,9,11,12,14
  
  
• WEEK 2 - January 22 - 26
  
  Mon. - 2.1: Complex Numbers.
  HW -- 2.1: 3,7,9,11,15,17,21,23,26,27 29,31,36
  (there is a misprint in 27; see errors in book above),
  
  Wed. - 2.1: Complex Numbers and problems 31 and 36.
  
  Fri. - 2.2: Complex Roots
  HW - 2.2: 1, 13, 23, 27, 19, 21, 28, 5, 8, 15 (do them in that order)
  NOTE: In #'s 19 and 21, it should be assumed that ``w'' is a cube root of unity
  Take-home Quiz 1 Given Out - Due on Wednesday
  
  
• WEEK 3 - January 29 - February 2
  
  Mon. 2.2: Finish Complex Roots.
  
  Wed. - 2.3: Solvability by Radicals and Algebraic Expressions
  HW -- 2.3: 1-17 (odd)
  
  Fri. - 2.4: Ruler and Compass Constructions
  HW - 2.4: 6,9, 11, 12, 13, 16, 19,21,23
  First Group Projects 2.4 # 1 and 5 assigned
  
  
• WEEK 4 - February 5 - February 9
  
  Mon. - 2.4: Finish Ruler and Compass Constructions
  
  Wed. - 2:5: Orders of roots of unity,
  Take-home Quiz 2 Given Out -- Due Monday
  HW - 2.5: 3,5,9,11,13,15,18
  
  Fri. - 3.1: The cubic Formula and 3.2: More on solvability by radicals
  Start 4.1: Modular Arithmetic
  HW - 3.1: 1,5,10,11
  
  
• WEEK 5 - February 12 - February 16
  
  Mon. - Answer questions on 2.4 and finish 4.1.
  HW - 4.1: 3, 6 and 7 (only with respect to #3), 9, 11, 15, 21 (21 was assigned in class but not on the web -- my error).
  
  Wed. - 4.2: The Euclidean Algorithm and start Modular Inverses
  4.2: 1, 2, 3, 5, 21, 22, 23, 25.
  
  Fri. 4.2: Finish Modular Inverses
  HW - 4.2: 11, 13, 17, 7, 15 (try in the above order)
  Next Group Projects 4.2: 24 and 26, 30.
  
  
• WEEK 6 - February 19 - February 23
  
  Mon. - 5.1: Start the binomial theorem
  Homework 5.1: 1,5,7,15.
  Take-home Quiz 3 Given Out -- Due Friday
  
  Wed. - 5.2: Fermat's Theorem
  HW - 5.2: 1,3,7,9,12.
  Take-home
  
  Fri. - Finish 5.2: orders of roots. Hopefully a group project
  HW - 5.2: 17, 22, 25, 27. solutions
  Practice Exam available.
  
  
• WEEK 7 - February 26 - March 2
  
  Mon. -- Some group projects, and review.
  
  Wed. -- In class exam.
  
  Fri. -- The handout on RSA (applied example), the handout on rings and fields and start 6.1: Fields and Polynomials.
  HW -- the problem of the RSA handout, problems 1-5 on the ring handout and 6.1: 5,7,18,21.
  
  
• WEEK 8 - March 5 - March 9
  
  Mon. The handout on rings and fields
  HW problems 1-5 on the ring handout and 6.1: 5,7,18,21.
  
  Wed. -- 6.1: Polynomials and Start 6.2: Factorization of Polynomials
  HW -- Problem 6 on the ring handout and 6.1: 1,3,4,11,22.
  
  Fri. -- Answer questions and tell bad jokes. NO Homework.
  
  

• Spring Break

• WEEK 9 - March 19 - March 23
  
  Mon. -- 6.2: Factorization of Polynomials
  HW -- 6.2: 3,5,10,12 (Hint: see #10), 13
  Take-home Quiz 4 Given Out -- Due Friday
  
  Wed. -- Finish 6.2 and Handout of irreducible polynomials in Q[x]
  HW -- 1-3 on the handout.
  
  Fri. -- 6.3: The Euclidean Algorithm (again)
  HW - 6.3: 21, 1, 3, 7, 8, 9 (hint look at the zeros of x^n-1).
  
  
• WEEK 10 -- March 26 - March 30
  
  Mon. -- 7.1: Galois Fields
  HW -- 7.1: 11,12,18-21,25.
  New group Projects 7.1: 26,29.
  
  Wed. - 7.1 and 7.2: Orders in Galois Fields:
  HW -- 7.1: 1,2,5 and 7.2: 4 (hint look at problem 7.1.5).
  
  Fri. - 7.2 and 7.3: More on orders:
  HW - 7.2: 1,3,10,11 and 7.3: 2,3,5 (hint see problem 7.1.2),
  7 (hint look at problem 7.1.5), 13 (where is a typo x^(p^v) should be x^(p^v-1)).
  Take-home Quiz 5 Given Out -- Due Wednesday.
  
  
• WEEK 11 - April 2 -- April 6
  
  Mon. Finish Chapter 7.
  
  Wed. - 8.2: Permutations
  HW -- Handout of problems for 8.2 Solutions
  
  Fri. - 8.2: Permutations II
  HW -- 8.2: 1, 3, 6, 8, 9, 11, 13, 22, 30, 31
  
  
• WEEK 12 - April 9 -- April 11
  
  Mon. - 8.4: Transposition and parity
  HW -- 8.4:1,3,7,9,11,12,13
  
  Wed. - Two group projects
  Answer lots of questions
  Take-home Quiz 6 Given Out -- Due Friday
  
  Fri. Easter break. No class.
  
  
• WEEK 13 - April 16 -- April 20
  
  Mon. - Easter break. No class
  
  Wed. - 9.1 and 9.2: Definition of a group, subgroup, order and examples.
  HW -- 9.1: 1-6, 35 and 9.2:1-4,7
  
  Fri. - 9.1 and 9.2: More examples of groups
  HW -- 9.2: 28, 37 and 9.3: 17, 19, 22
  
  
• WEEK 14 - April 23 -- April 27
  
  Mon. - Catch up and review.
  
  Wed. - In class exam.
  
  Fri. - 9.4: Subgroups and cosets
  Group Project 9.5 #12 and #28.
  Homework - 9.4: 1, 2, 6, 11, 13, 15, 28, 31, 32, 33 (should be 12 not 15), 36, 41, 56 (HW need not be turned in)
  Last day to turn in HW
  
  
• WEEK 15 - April 30 -- May 2
  
  Mon. - 9.5: Cyclic groups
  Homework - 9.5: (for the first 4 problems just decide if it is cyclic or not) 2, 3, 4, 10, 13, 15, 16, 17 (HW need not be turned in).
  
  Wed. Last group project, catch up and review.
  
  
• Final Exam, Monday May 7, 1:45 -- 3:45, FITZ 356.
  
  
• Material not covered this year.
  
  9.3: Isomorphisms of groups
  HW -- 9.3: 2, 3 (hint: write out the tables and compare) 9-16
  
  Parts of 10.1: Definition of normal subgroups, Examples S_3 and S_5 and Prop 10.11.
  Homework - 10.1: 7,9 (only the first part: is H normal in G?), 17,22,36
  
  Isomorphisms of rings, characteristic 0 vs p, examples x^3-2 and x^5-1 over the rationals, Galois's theorem about solvable by radicals, the example x^5 - 10x +5 over the rationals (For interested parties, I suggest reading the handout, Introduction to Galois Groups).
  
  

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