MAT 347: Groups, Rings, and Fields

Shortcuts:

• Announcements.
• Part 1: Group theory.
• Part 2: Ring theory.
• Part 3: Fields and Galois theory.

ANNOUNCEMENTS

• Course outline
• Florian's office hours are Tuesdays, 3-5pm in BA 6186.
• First term test will be held on Wednesday, November 21, 3:10-5:00pm at EX300.
• Second term test will be held on Wednesday, February 13, 3:10-5:00pm (location was sent by e-mail). [Reschedule due to school closure on Feb 6.]
• The final exam will be held on Monday, April 22, 9:00am-12:00pm 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

Second term test

Final exam

HOMEWORK ASSIGNMENTS AND CALENDAR

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 1-page summary. Please make sure that you are comfortable with equivalence relations and partitions (that they are equivalent 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 2-subgroup 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 make-up class on Thu Dec 6, 3-4pm 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 non-zero ring has a maximal ideal.
  
  
• Mon Jan 7 -- Fri Jan 11
  • Read section 7.6, and either of:
    • Alfonso's notes on Unique Factorization. I will follow these notes in class!
    • Sections 8.3, 8.2, 8.1 in the book. Note that order and emphasis is a bit different from the notes.
  • 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(1-e) 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:
    • Alfonso's notes on Unique Factorization. I will follow these notes in class!
    • Sections 8.3, 8.2, 8.1 in the book. Note that order and emphasis is a bit different from the notes.
  • 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 F₂ 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 non-zero non-unit is a product of prime elements.
    • [Problem 2] In Z[i] let a = 47-13i 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 271-272 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 non-zero. 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 1-2 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 2-3 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 Mar 1
  • Read Sections 3-4 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 n-gon 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 4-5 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 (1811-1832).
  
  
• Mon Mar 11 -- Fri Mar 15
  • Read Sections 5-6 of Alfonso's notes OR Sections 13.4, 13.5 (book).
  • Worksheet from March 12: Splitting fields and normal extensions.
  • Homework #18 (due Friday March 22):
    • Section 13.4: 1, [2] (careful when calculating the degree!), 3, 4, 6
    • [Problems 1-3] in this Handout
  • You can now understand an "algebraic proof" of the Fundamental Theorem of Algebra here. It uses the fact that any symmetric polynomial can be expressed in terms of the elementary symmetric polynomials. There's a perhaps easier "algebraic proof" using Galois theory at the same link (or in sec 14.6 of the book), and you will be able to understand it soon.
  
  
• Mon Mar 18 -- Fri Mar 22
  • Read Sections 6-7 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, 563-566, 576-581.
  • 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 1-3] 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.