MAT 347: Groups, Rings, and Fields
This is the official website of the course MAT347 at the University of Toronto in the academic year 20152016.
Shortcuts:
ANNOUNCEMENTS
 Course outline
 Joel's office hours are Thursdays 35 pm in BA 6110. I am also available at other times by appointment.
 Jonathan's office hours are Tuesdays 121 pm in HU 1018.
 Online forum: we will be using Piazza as an online forum for discussions among students. The professor and TA will participate in the discussions as well. So we encourage you to post questions there.
 First midterm will be held on Friday November 20, 10:15  12:00, in EX 300.
 Guest lecture on the Rubik's cube by Peter Tingley (Loyola), Monday November 23.
 Second midterm will be held on Friday February 5, 10:15  12:00, in BA 2175.
Final exam
The final exam will be on April 13, 2:005:00 in the St. Vladimir Institute (really). The format of the exam will be similar to the midterms.
The exam will cover the material from the whole course, though the emphasis will be on the more recent material. You should review all the proofs from the Galois theory part of the course. It would be a good exercise to try to fill in all the proofs in Alfonso's notes.
Jonathan will hold a review session on Friday April 8, 10:0012:00 (the last day of classes). At the review session, Jonathan will distribute a list of review problems (we will also post it to the website).
I will have office hours on Monday April 11, 10:0012:00 and Tuesday April 12, 2:005:00.
Jonathan will have office hours on Monday April 11, 4:305:30 and Tuesday April 12, 10:0012:00.
Here are some review problems for the final.
Second midterm
The second midterm will be held on Friday February 5, 10:1512:00 in BA 2175. The only material from group theory will be finitely generated abelian groups. There will also be all the ring theory material, up to and including irreducibility criteria. So you are responsible for all lecture material and textbook chapters 5.2, 7, 8, 9 (except 9.6).
The format will be similar to the first midterm in terms of True/False, definitions, and some longer answer questions. I suggest that you study by reviewing all the homework problems (including the ones you didn't have to hand in).
First midterm
The first midterm will be held on Friday November 20, 10:1512:00 in EX 300. It will cover all the material up to and including semidirect products. So you responsible for all material from lectures, textbook chapters 1  5 (except 5.2), and 6.3. You are also responsible for homework assignments 18.
I suggest that you study by reviewing your notes, reading the textbook, and going over the homework assignments. You might also find it helpful to practice nonassigned questions from the textbook. You should focus your studying on the "core" material from the course: group actions, subgroups, quotient groups, symmetric group.
There will be some true/false questions, some definitions, and longer answer questions. For the true/false questions, you will need to think about groups having certain properties.
HOMEWORK ASSIGNMENTS AND CALENDAR
I encourage you to attempt the reading assignments before the lectures on that topic start.
I will post every homework assignment here at least one week before it
is due. I will not update them without warning less than a week before
they are due.
I expect you to do all the problems in the homework set, but only the
ones in bold and brackets are to be turned in on the day the homework
set is due. Sometimes the nottobehandedin problems will help you
solve the tobehandedin problems. They are due at the beginning of
the class.
I will not accept late assignments.
Most of these worksheets (and this webpage design) were graciously donated by Alfonso GraciaSaz.
PART 1: Group theory.
 Mon Sept 14  Fri Sept 18
 Reading:
 Section 0.1.  These are basic concepts that you should know before the course starts, and that I will not cover in lecture.
 Section 0.2.  I will come back to cover this later in the course, but it will still be beneficial to read it now.
 Sections 0.3, 1.1, 1.2, 1.3, 1.4.
 Worksheet from Sept 18: Order
 Homework #1 (due on Friday, September 25):
 Section 1.1: problems 6, 14, 15, [20], 22, [25], 31.
 Section 1.2: problems 1, 3, [5], [10] , 13, 15.
 Section 1.3: problems 1, 9, 15.
 Section 1.4: problem [2].
 Marking scheme for homework #1.
 Some extra links for those interested in defining groups via presentations:
 If you want to see an example of a group presentation whose word problem is undecidable, see this paper. The example given has 10 generators and 27 relations.
 If you are curious to learn more about how (computationally)
hard it is to understand abstract groups given by generators and
relations, you can read the first section of this paper.
This may not make sense unless you have an interest in computer science or in logic and foundations.
 Mon Sept 21  Fri Sept 25
 Read section 1.5, 1.6, 1.7, 2.1
 Worksheet from Sept 21: Presentations
 Worksheet from Sept 25 and 28: Counting, group actions, and the OrbitStabilizer Lemma
 Homework #2 (due on Friday October 2):
 Section 1.4: 3, 11.
 Section 1.5: [1]
 Section 1.6: 1, 2, 4, 6, 7, 10, [17], 18, 19, 20.
 Section 1.7: 11, 14, 15, [17], [21], 23.
 Marking scheme for homework #2.
 In class, we discussed a presentation of the Dihedral group with two generators, each of which squared to 1. This is a special case of a Coxeter group, named after Toronto's most famous mathematician H.S.M. Coxeter.
 Mon Sept 28  Fri Oct 2
 Read sections 4.1, 2.1, 2.2, 2.3 and these notes on Burnside's Lemma.
 Worksheet from October 2: Cyclic groups and cyclic subgroups.
 Homework #3 (due on Friday October 9):
 Section 1.7: 18, [19]
 Section 2.1: 2, [4], 6, 8, [9], 10
 Section 2.3: 2, 10, [26]
 Burnside Lemma handout: [2], 4
 Marking scheme for homework #3.
 Mon Oct 5  Fri Oct 9
 Read sections 2.4, 2.5, 3.1, 3.2
 Homework #4 (due Friday October 16):
 Worksheet from October 7: Quotient groups.
 Worksheet from October 9: Joins.
 Section 2.2: 6, 7, [10],14
 Section 2.4: 5, [7], 8, 14
 Section 3.1: 1, 3, 9, 22, [32], 34,[35]
 Section 3.2: 5, 8, [11]
 Marking scheme for homework #4.
 Here is a nice picture of the lattice of subgroups of the symmetric group S_4.
 Wed Oct 14  Fri Oct 16
 Read sections 3.3, 3.4, 3.5
 Worksheet from October 16: The symmetric and the alternating groups
 Homework #5 (due Friday October 23):
 Section 3.1: 11, [12], 13, 32, 35, [36], 41
 Section 3.2: [16], 20, 22, 23
 Section 3.3: 2, 3, [5], 7
 Section 3.5: 2, 3, 4, [7]
 Marking scheme for homework #5.
 Mon Oct 19  Fri Oct 23
 Read sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
 Worksheet from October 23: The action of a group on itself by conjugation.
 Homework #6 (due Friday October 30):
 Section 3.4: 5, [6], 7, 9
 Section 3.5: 9, 12, [16], 17
 Section 4.1: 2, [3], 4, 6, 8
 Section 4.2: 4, [7], 8
 Section 4.3: 2, 3, 4, 8, [22]
 Marking scheme for homework #6.
 If you are interested you can read about the monster, the largest sporadic finite simple group.
 Mon Oct 26  Fri Oct 30
 Read sections 4.4, 4.5, 5.1, 6.3
 Worksheet from October 30: Free groups.
 Homework #7 (due Friday November 6):
 Section 4.3: 23, [24], 25, [32]
 Section 4.4: 6, 7, [8], 10, 18, 19
 Section 4.5: 4, [9], 10, 21
 Section 5.1: 1, 2, [5], 8
 Marking scheme for homework #7.
 Mon Nov 2  Wed Nov 11
 Read sections 5.1, 5.4, 5.5, 6.3
 Worksheet from November 6: Semidirect products.
 Homework #8 (due Friday November 13):
 Section 5.1: [12], 13, 14, 15, 17
 Section 5.4: [5], 11, [14],16, [19], 20
 Section 5.5: 1, 2, 6, 9, 10, 19, [22]
 Marking scheme for homework #8.
 Fri Nov 13  Wed Nov 18
PART II: Ring theory.
 Wed Nov 25  Fri Nov 27
 Read sections 7.1, 7.2, 7.3.
 Worksheet from November 27: Subrings, ideals, and ring homomorphisms
 Homework #10 (due Friday December 4):
 Section 7.1: 1, 3, 4, 6, 13, [14], 21, [25], 26
 Section 7.2: 2, 3, 5
 Section 7.3: 1, 5, 6, [12], 15, [21]
 Marking scheme for homework #10.
 Mon Nov 30  Mon Dec 7
 Read sections 7.4, 7.5.
 Worksheet from December 4: Fields of fractions
 Homework #11 (due Friday January 15):
 Section 7.3: 24, 26, [28], [29], 31, 36
 Section 7.4: [7], [9], 11, 15, 19, 26, [31], 33, 37
 Marking scheme for homework #11.
 Mon Jan 11  Fri Jan 15

Read sections 7.6, 8.1, 8.2, 8.3 and these notes on Unique Factorization Domains
 Worksheet from January 15: Factorization, GCDs, and Ideals
 Homework #12 (due Friday January 22):
 Section 7.4: 14, [15], 25, 40
 Section 7.6: 1, 3, [4], 7
 Notes on UFDs, Exercises: 1, 2, 3, [4], 5, [6], [7]
 Marking scheme for homework #12.
 Mon Jan 18  Fri Jan 22
 Read sections 8.1, 8.2, 8.3 and the notes of UFDs.
 Worksheet from January 22: Factorization in the Gaussian integers
 Homework #13 (due Friday January 29):
 Section 8.2: 2, [3], [5], 8
 Section 8.3: 2, 3, 5, 8
 Notes on UFDs, Exercises: 8, 9, 10, [12], 13, [14], 15
 Marking scheme for homework #13.
 In class we discussed the problem of determining which quadratic integer rings were PIDs. For more history on this problem, including connections with number theory, take look at this paper.
 Mon Jan 25  Fri Jan 29
PART III: Fields and Galois theory.
 Mon Feb 1  Wed Feb 3
 Read sections 9.5, 13.1, 13.2, sections 1, 2 of Galois theory notes
 Homework #14 (due Friday February 12):
 Section 9.1: 4, 5, 6, 8, 13, 18
 Section 9.2: 1, [2], 4, 5, 9, 12
 Section 9.3: 1, 2, 3, [4]
 Section 9.4: 2, 3, [4], [7], 8, 9, 10, 12, 14, 18
 Section 13.1: [1], 2, 5
 Marking scheme for Homework #14.
 Mon Feb 8  Fri Feb 12
 Read sections 13.2, 13.3 of textbook, sections 2, 3 of the notes.
 Worksheet from February 12: Constructions with straightedge and compass.
 Homework #15 (due Friday February 26):
 Section 13.2: 3, 4, 5, [7], 10, [13], [14], 16, [18], 19, 20, 21
 Marking scheme for Homework #15.
 Mon Feb 22  Fri Feb 26
 Read sections 4 and 5 in the notes, sections 13.4 and 14.1 in the textbook.
 Worksheet from February 26: The Galois correspondence
 Homework #16 (due Friday March 4):
 Hand in all problems from this handout
 Section 14.1: 1, 2, 3, 4, 5, 7, 10
 Marking scheme for Homework #16.
 Mon Feb 29  Fri Mar 4
 Read section 5 and 6 in the notes, sections 13.4 and 13.5 in the textbook.
 Worksheet from March 4: Separable extensions
 No homework this week.
 Mon Mar 7  Wed Mar 9
 Read sections 5 and 6 in the notes, section 13.4, 13.5, 14.1 in the textbook.
 Homework #17 (due Friday March 18):
 Section 13.4: 1, [2], 3, 4, [5], 6
 Section 13.5: 1, 2, 8
 Section 14.2: 3, 4, [7], [13], [14]
 Marking scheme for Homework #17.
 Mon Mar 14  Fri Mar 18
 Read section 7 in the notes, section 14.2 in the textbook.
 Worksheet from March 18: An example of the FTGT
 Homework #18 (due Monday March 28):
 Hand in all problems from this handout.
 Section 14.2: 10, 11, 12, 16, [17], 18, 29, 31
 Marking scheme for Homework #18.
 Mon Mar 21  Wed Mar 23
 Read sections 7 and 8 in the notes, sections 13.6, 14.2 and 14.5 in the textbook.
 Homework #19 (due Monday April 4):
 Section 13.6: 1, 6, 9, 10
 Section 14.5: 1, 3, 4, [5], 7, [8], [10], 11
 Section 14.6: 2, 4, 10, [13], 17
 Marking scheme for Homework #19.
 Mon Mar 28  Fri Apr 1
 Read section 10 in the notes and/or 14.6, 14.7 in the textbook.
 Worksheet from April 1: Computing Galois groups.
 There is a very fascinating history related to solving polynomial equations and Galois theory. Here is the biography of Galois. On the same site, you can find the biography of Tartaglia, the person who solved the cubic equation.