MAT 347: Groups, Rings, and Fields
This is the official website of the course MAT347 at the University of Toronto in the academic year 20182019.
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ANNOUNCEMENTS
 Course outline
 Florian's office hours are Tuesdays, 35pm in BA 6186.
 First term test will be held on Wednesday, November 21, 3:105:00pm at EX300.
 Second term test will be held on Wednesday, February 13, 3:105:00pm (location was sent by email). [Reschedule due to school closure on Feb 6.]
 The final exam will be held on Monday, April 22, 9:00am12:00pm at Cartwright Hall, St. Hilda's College, 44 Devonshire Place.
 Tutorials start in Week 2 (September 18).
 Errata for Dummit and Foote (3rd ed), by Richard Foote
(some may already be incorporated, depending on when your book was printed)
First term test
The first midterm will be held on Wednesday, November 21, 3:105:00pm at EX300. It will cover all the material up to and including semidirect products. You are responsible for all material from lectures, textbook chapters 15 (except 5.2, 5.3). 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.
No materials (textbook, notes,...) will be allowed.
You do not need to memorise proofs from class, in other words I would not ask you to prove one of the difficult theorems from class (such as Sylow theorems or simplicity of A_n).
Second term test
The second midterm will be held on Wednesday, February 13, 3:105:00pm. It will cover all the material that we have covered that was not on the first term test. You are responsible for all material from lectures, textbook chapters 5.2 (finite abelian groups), 6.3 (free groups), 79 (ring theory; except 9.6), and Alfonso's notes on UFDs. You are also responsible for homework assignments 914.
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.
No materials (textbook, notes,...) will be allowed.
You do not need to memorise proofs from class, in other words I would not ask you to prove one of the difficult theorems from class.
Final exam
The final exam will be held on Monday, April 22, 9:00am12:00pm. It will cover all the material that we have covered in this course.
The exam will cover the material from the whole course, though the emphasis will be on the recent material.
(I'd say just over half will be on fields and Galois theory.)
I will have office hours Tue April 16, 46pm and Thu April 18, 46pm. I will be available in the week of April 8, if needed.
Daniel will have office hours Thu April 18, 911am at BA6180.
I suggest that you study by reviewing your notes, reading the textbook as well as Alfonso's notes (UFDs and Galois Theory), and going over the homework assignments. You might also find it helpful to practice nonassigned questions from the textbook.
There are also some review problems for the final (by Jonathan Love).
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.
You can certainly discuss homework problems with other students, but you need
to write your solutions in your own words. If you find the solutions in e.g. books
or the internet you must quote your sources.
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 10  Mon Sept 17
 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 12: Order
 Homework #1 (due on Friday, September 21):
 Section 0.3: problem 13.
 Section 1.1: problems 6, 9, 14, [21], 22, [25], 31.
 Section 1.2: problems 1, 3, [4], 9, 12, 15.
 Section 1.3: problems 2, 11, 15.
 Section 1.4: problem [2].
 [Problem]: For every positive integer n, find how many elements of order n there are in S_5.
 Why groups? (by Keith Conrad)
 Cayley's 1854 paper in
which he introduces finite groups, see the top of p.41 (though he thinks of elements as operations, so his "axioms" look a little different from ours, e.g. associativity is automatic for him). He also classifies groups of order 4 and 6.
 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 17  Fri Sept 21
 Read sections 1.5, 1.6, 1.7, 2.1.
 Worksheet from Sept 18: Presentations
 Worksheet from Sept 19: Group actions
 Homework #2 (due on Friday September 28):
 Section 1.5: [1]
 Section 1.6: 1, 2, 4, 6, 7, 14, 18, [20], 23, 26.
Problem 26 gives a nicer way to check the multiplication in Q8 is associative! To do that, show that if you have (1) a group (G,*), (2) a set H with binary operation #, having identity and inverses,
and (3) an injective map f: H > G that satisfies the homomorphism property (i.e. f(x#y) = f(x)*f(y) for all x,y), then # is associative and hence H is a group.
 Section 1.7: 3, 5, 8, 11, [17], 18, [21].
(We'll define subgroups and kernels on Monday next week.)
 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.
(The same is true for the infinite dihedral group I mentioned.)
 Mon Sept 24  Fri Sept 28
 Read sections 4.1, 2.1, 2.2, 2.3.
 Worksheet from Sept 25: Orbits and Stabilizers.
 Worksheet from Sept 26: Cyclic groups.
 Homework #3 (due on Friday October 5):
 Section 2.1: 3, 8, 9, [12], 14 (optional: 16)
 Section 2.2: 4 (you may assume Lagrange), 7, [10], 12 (optional: 14)
 Section 2.3: 3, 10, [11], 12, 18, [26] (optional: 1, 9, 24)
 Mon Oct 1  Fri Oct 5
 Read sections 2.4, 2.5, 3.1, 3.2.
 Worksheet from October 2: Joins and Quotient groups.
 Worksheet from October 3: Quotient groups.
 Homework #4 (due Friday October 12):
 Section 2.4: 7, 13, [14] (optional: 11)
 Section 2.5: 8, 10
 Section 3.1: 5, 14, 16, 17, [24], [32], 36, 40 (optional: 19, 31, 32, 37, 41, 42)
 Section 3.2: 5, 8, 12 (G does not have to be finite!), [16]
 Here is a nice picture of the lattice of subgroups of the symmetric group S_4.
 Equivalence relations: here is a nice 1page summary. Please make sure that you are comfortable with
equivalence relations and partitions (that they are equivalence concepts).
 Tue Oct 9  Fri Oct 12
 Read sections 3.3, 3.4, 3.5.
 Worksheet from October 9: The symmetric and the alternating groups.
 Homework #5 (due Friday October 19):
 Section 3.3: 2, 4, 7, [9].
 Section 3.5: 3, [4], 5, 9, 10, 12.
 [Problem 1] Suppose that H, K are normal subgroups of G such that their intersection is {1} and HK = G. Show
that G is isomorphic to the direct product H x K. (Hint: show that hk = kh for h in H, k in K, by showing
that hkh^{1}k^{1} is contained in both H and K.)
 [Problem 2] Suppose that f : G > H is a group homomorphism and H is abelian. If K is any subgroup of G
that contains ker(f), show that K is normal in G.
 [Problem 3] We say that a group G is almost abelian if it has a normal subgroup N such that
both N and G/N are abelian. (For example, S_3 is almost abelian. Why?) Show that any subgroup of an almost abelian group is almost abelian.
 Mon Oct 15  Fri Oct 19
 Read sections 4.1, 4.2, 4.3, 4.4.
 Worksheet from October 16: The action of a group on itself by conjugation.
 Homework #6 (due Friday October 26):
 Section 3.4: [5], 6, 7, 8 (optional: 9).
 Section 4.1: [1], 2, 3, 6, [9].
 Section 4.2: [7], 8, 10.
 Section 4.3: 2, 4, 9, [31] (optional: 3, 22).
 As promised, here are some notes on Burnside's Lemma which allows you to count the number of orbits of a group action.
 If you are interested you can read about the monster, the largest sporadic finite simple group. See also here.
 Mon Oct 22  Fri Oct 26
 Read sections 4.4, 4.5, 5.1.
 Worksheet for this week: Proof of the Sylow Theorems.
 Homework #7 (due Friday November 2):
 Section 4.3: 29.
 Section 4.4: 1, 3, 6 (characteristic is defined on p.135), [8], [14] (optional: 4, 7, 18, 19).
 Section 4.5: 3, 5, [13], 24, [31] (for A_5 only, plus give one example of a Sylow 2subgroup for S_5), [32], 45 (optional: 17, 46).
 Mon Oct 29  Fri Nov 2
 Read sections 5.1, 5.4, 5.5, 6.3.
 Worksheet from October 30: Semidirect products.
 Homework #8 (due Friday November 16):
 [Problem 1]
 (a) Suppose that H, K are subgroups of a group G such that HK = G and hk = kh for
all h in H and k in K. Prove that G is isomorphic to (H x K)/Z for some normal subgroup Z
isomorphic to the intersection of H and K. Show moreover that H and K are normal in G and that the
intersection of H and K is contained in Z(G).
 (b) Given groups A, B and subgroups Z1 of Z(A) and Z2 of Z(B) and an isomorphism f : Z1 > Z2.
Define A * B := (A x B)/N, where N = {(z, f(z)^{1}) : z in Z1} (check it's normal!). Show that A * B has subgroups
H, K as in the first sentence of part (a) that are isomorphic to A, B (respectively).
(See crowdmark for better formatting...)
 Section 5.1: 1, 4, 14 (optional: 15, 17)
 Section 5.4: [4], 5, 7, 15, 16, [17], 19, 20
 Section 5.5: 1, 2, 6, [7bcd], 18, [22]
 Mon Nov 12  Fri Nov 16
 Read sections 6.3, 5.2.
 Worksheet from November 13: Classification of finite abelian groups.
 Remember that we have Test 1 next week! Please click the link to the left for more information
(see also my announcement on Quercus).
 Mon Nov 19  Mon Nov 26
 Read section 5.2.
 Homework #9 (due Friday November 30):
 Section 6.3: 4, [5], 7, [9], [10], 11, [14] (see crowdmark if your book doesn't contain the last problem)
 Section 5.2: [8] (you can ignore the hint in part (a))
 A fun song on group theory, by Northwestern grad students:
"Finite simple group of order two".
PART 2: Ring theory.

Important: List of differences between my conventions and Dummit&Foote (updated Dec 6)
 Tue Nov 27  Fri Nov 30
 Read sections 7.1, 7.2, 7.3.
 Worksheet from November 27: Subrings, ideals, and ring homomorphisms.
 Homework #10 (due Friday January 11)  NOTE: same due date as #11:
 Section 5.2: 3e, 4, [5] (plus: find all abelian groups of order 10000 with exponent 100), 12 (optional: 2, 14)
 Section 7.1: 4, [7], 8, [14], 15 (optional: 11, 21, 23, 25)
 Section 7.2: 2, 3, 12 (optional: 6, 7, 8, 13)
 Section 7.3: [5], 8, 19, [24] (optional: 10)
 Mon Dec 3  Fri Dec 7
 Read sections 7.4, 7.5, Appendix I.2.
 Worksheet from December 4: Fields of fractions.
 We will have a makeup class on Thu Dec 6, 34pm at GB220 (next to our usual class room).
 I will not have regular office hours during exam period, but let me know if you want to meet.
 Homework #11 (due Friday January 11)  NOTE: same due date as #10:
 Section 7.3: 26, [34]
 Section 7.4: 2, 7, [8], 9, [13] (with my conventions, in (a) show that phi^{1}(P) is prime!), 15 (see hint for 14a)
 Section 7.5: [3], 4
 More on Zorn's lemma (by Keith Conrad)
 For fun: a rap version of the proof that every nonzero ring has a maximal ideal.
 Mon Jan 7  Fri Jan 11
 Read section 7.6, and either of:
 Worksheet from January 8: Factorization, GCDs, and Ideals.
 Homework #12 (due Friday January 18):
 Section 7.4: 19 (use a result from Section 7.1), [25], 26, [30] (note: the first part implies Ex. 7.3/29! why?)
 Section 7.6: [1] (note: Re, R(1e) are not subrings with my conventions!), 3, 4, 7
 Alfonso's notes: 1, 2, 3, [4], [6], 7
 Example of a Bezout domain that is not a PID.
 Mon Jan 14  Fri Jan 18
 Read either of:
 Worksheet from January 15: Factorization in the Gaussian integers.
 Homework #13 (due Friday January 25):
 Alfonso's notes: 8, 10, 12, [14] (also over the field F_{2} of two elements), 15
 [Problem 1] Show that any irreducible element in a UFD is prime. Deduce that an integral domain is a UFD
iff every nonzero nonunit is a product of prime elements.
 [Problem 2] In Z[i] let a = 4713i and b = 53+56i. Find a generator d of the ideal (a,b). Also, find x,y in Z[i] such
that d = ax+by. (Hint: use the Euclidean algorithm in Z[i]. This is described in Example 3 on pages 271272 of the book.)
 Section 8.2: 2, [3], 5, 6
 Section 8.3: 2, 3, [6], 8 (optional: 5, 11)
 For an example of a PID that is not Euclidean, see the Example on p.277 of our book.
 For an example of a UFD that is not PID, see Example 3 on p.285 of our book (we'll get there next week).
 Next week we'll also discuss an example of an irreducible element (in some integral domain) that is not prime.
 Mon Jan 21  Fri Jan 25
 Read Sections 9.1, 9.2, 9.3, 9.4.
 Worksheet from January 21: Factorization in polynomial rings.
 Homework #14 (due Friday February 1):
 Section 9.1: 5, [6], 13
 Section 9.2: [1], 2, 5, [11] (optional: 7)
 Section 9.3: [1], 2, 4 (optional: 5)
 [Problem] Suppose that R is a UFD and that f, g in R[x] are nonzero. Show that f  g in R[x] if and only if (f  g in F[x] and C_f  C_g in R). Here, C_f denotes the content of f, as defined in the worksheet.
 Mon Jan 28  Fri Feb 1
 Read Sections 9.4, 9.5, 13.1 (book), Sections 12 of Alfonso's Galois theory notes.
 Worksheet from January 29: Irreducibility criteria.
 Remember that we have Test 2 next week! Please click the link to the left for more information
(see also my announcement on Quercus).
PART 3: Fields and Galois theory.

Important: I will mostly follow Alfonso's Galois theory notes (last updated Apr 21).
Please let me know if you find any typos!
 Mon Feb 4  Fri Feb 8
 Read Sections 13.1, 13.2 (book), Section 2 of Alfonso's Galois theory notes (posted just above).
 Homework #15 (due Friday February 15):
 Section 9.4: 1, 2, [3] (Hint: see crowdmark), 5, [6], 9, 11, [13], 14, 17
 Section 9.5: [2] (parts a, c only), 3, 4
 Section 13.1: [1] (express your answer as a linear combination of 1, theta, theta^2), 2, 5, 8
 Mon Feb 11  Fri Feb 15
 Read Sections 23 of Alfonso's notes OR Sections 13.2, 13.3 (book).
 This week we'll have our rescheduled Test 2, but please don't forget about Homework #15 posted last week.
 Worksheet from February 12: Constructions with straightedge and compass.
 No homework this week, due to Test 2.
 Mon Feb 25  Fri Feb 29
 Read Sections 34 of Alfonso's notes OR Sections 13.3, some of 14.1 (book).
 Worksheet from February 26: Constructions with straightedge and compass. (continued)
 Homework #16 (due Friday March 8):
 Section 13.2: 1, 4, [5] (can you use the tower law?), [7], 8, 9, 12, 14, [16], 17, [18], 20
 Section 13.3: 1, 5
 [Problem]: Let P denote the positive integers n such that the regular ngon is
constructible with straightedge and compass.
(a) Prove that n in P implies 2^k n in P (for any k \ge 0).
(b) Prove that (n in P and m divides n) implies that m in P.
(c) Prove that if m, n are relatively prime and m, n both in P, then mn in P.
(Hint: use Bezout's identity to relate 2pi/mn, 2pi/m, and 2pi/n.)
 If you are interested, here are short (but clever!) proofs that e and pi are transcendental.
 Mon Mar 4  Fri Mar 8
 Read Sections 45 of Alfonso's notes OR Sections 13.4, 14.1 (book).
 Worksheet from March 5: The Galois group.
 Homework #17 (due Friday March 15):
 Section 14.1/[7], 9 (this is easier if char k is 0), 10
 [Problems] in this Handout [Update: apologies, in problem 3(e) I meant the *lower* triangular unipotent matrices!]
 Here or here is the biography of Galois (18111832).
 Mon Mar 11  Fri Mar 15
 Mon Mar 18  Fri Mar 22
 Read Sections 67 of Alfonso's notes OR Sections 13.5, 14.2 (book), but note the book covers less about separable extensions.
 Worksheet from March 19: An example of the FTGT.
 Homework #19 (due Friday March 29):
 Section 13.5: 1, 5, 11
 Section 14.2: 1, [3], 4, 5, 7, [11], 12, [13], 14, 15, [16]
(Hint: to compute the degree for 14.2/3, rule out that root(5) is in Q(root(2),root(3)), for example by first considering
the possible quadratic subfields of Q(root(2),root(3)).)
 Optional (but great for practice!): determine the lattice of intermediate fields in Problem 14.2/16.
Which of them are normal over Q?
 Mon Mar 25  Fri Mar 29
 Read Sections 7, 9, beginning of 10 of Alfonso's notes OR Sections 14.2, 14.3, beginning of 14.6/14.7 (book).
 Read all Examples in the book about the FTGT: book, pages 559, 560, 563566, 576581.
 Worksheet from March 26: Finite fields.
 Homework #20 (due Friday April 12):
 Section 14.3: 4, [5], 8, 9, 11
 Section 14.4: 1, 2
 Section 14.5: 1, 3
 [Problems 13] in this Handout
 Mon Apr 1  Fri Apr 5
 Read Sections 10 and 8 of Alfonso's notes OR Sections 14.6/14.7, 14.5 (book).
 Worksheet from April 2: Computing Galois groups.