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
• Office hours are Thursdays 3:30-4:30 or by appointment at BA 6186.
• First term test will be held on Monday, November 18, 3:10-5:00pm at EX310.
• Second term test will be held on Monday, February 3, 3:10-5:00pm at EX100.
• The final exam will be held on Thursday, April 23, 2:00pm-5:00pm at GB304.
• Tutorials start on September 10. They are an important part of the course!
• 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 9 -- Mon Sept 16
  • Reading:
    • Section 0.1. - These are basic concepts that you should know before the course starts, and that I will not cover in lecture. (You may not have seen the important concept of an equivalence relation before. We will review it in class.)
    • 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.
  • Worksheet from Sept 10: Order
  • Homework #1 (due on Sunday, September 22):
    • Section 0.1: problem 7.
    • Section 0.2: problem 1, 2, 3.
    • Section 0.3: problem 4, 7, [12], [13].
    • Section 1.1: problems 1, 6, [9], 18, 21, [23], [25], 31, 36.
  • Equivalence relations (see the first three sections; these notes go into slightly more detail than the textbook)
  • 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.
  
  
• Mon Sept 16 -- Fri Sept 20
  • Read sections 1.2, 1.3, 1.4, 1.5, 1.6, 1.7.
  • Worksheet from Sept 17: Presentations
  • Homework #2 (due on Sunday September 29):
    • Section 1.2: 1, [5], 7 (optional: 4, 9, 17, 18).
    • Section 1.3: 3, [4] (it's ok to list elements by type, but please explain how many elements there are of each type), [14] (please replace "commuting p-cycles" by "disjoint p-cycles") (optional: 2, 9, 11).
    • Section 1.4: 2, 8 (optional: 10, 11).
    • Section 1.5: 1 (optional: 2)
    • Section 1.6: 1, [3], 4, [17], 20 (optional: 5, 6, 11, 18, 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.
  • 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 23 -- Fri Sept 27
  • Read sections 1.7, 4.1, 2.1, 2.2, 2.3.
  • Worksheet from Sept 23: Group actions
  • Worksheet from Sept 24: Orbits and Stabilizers, Cyclic groups.
  • Homework #3 (due on Sunday October 6):
    • Section 1.7: 3, 6, 9, 11, 17, [18] (optional: 8, 11, 21)
    • Section 2.1: 3, 6, [8], 11, 15 (optional: 16)
    • Section 2.2: 4 (you may assume Lagrange), 7, [10], 12 (optional: 14)
    • Section 2.3: 4, 11, 18, [26] (optional: 1, 9, 24)
  
  
• Mon Sept 30 -- Fri Oct 4
  • Read sections 2.4, 2.5, 3.1, 3.2.
  • Worksheet from September 30: Quotient groups.
  • Homework #4 (due Sunday October 13):
    • Section 2.4: 6, 13, 14 (optional: 15, 16)
    • Section 2.5: 8, 10 (optional: 6, 9)
    • Section 3.1: 5, 16, 17, 24, [32], [36], 40 (optional: 14, 19, 31, 37, 41, 42)
    • Section 3.2: 4, [5], 8, 12 (G does not have to be finite!), [16] (optional: 9, 11)
    • [Problem] For each of the following statements, either give a proof or a counterexample.
      (i) If K is a subgroup of H and H a subgroup of G, then K is a subgroup of G.
      (ii) If K is a normal subgroup of H and H a normal subgroup of G, then K is a normal subgroup of G.
  • Here is a nice picture of the lattice of subgroups of the symmetric group S_4.
  
  
• Mon Oct 7 -- Fri Oct 11
  • Read sections 3.3, 3.4, 3.5.
  • Worksheet from October 7: The symmetric and the alternating groups.
  • Homework #5 (due Sunday October 20):
    • Section 3.3: 2, 3, [7], 9.
    • Section 3.5: 3, 4, [5], 9, 10, 12 (optional: 7, 17).
    • [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.
  • Isomorphism theorems for vector spaces (if you are interested)
  
  
• Mon Oct 14 -- Fri Oct 18
  • Read sections 4.1, 4.2, 4.3.
  • Worksheet from October 15: The action of a group on itself by conjugation.
  • Homework #6 (due Sunday October 27):
    • Section 3.4: 5, [6], 7, 8 (optional: 1, 9).
    • Section 4.1: 1, 2, [3], 7, 8 (optional: 4, 6).
    • Section 4.2: 7, 8, [11] (optional: 6, 9).
    • Section 4.3: 2, 4, 9, [22], 29 (optional: 3).
  • 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 21 -- Fri Oct 25
  • Read sections 4.4, 4.5, 5.1.
  • Worksheet for this week: Proof of the Sylow Theorems.
  • Homework #7 (due Sunday November 3):
    • Section 4.3: 32.
    • Section 4.4: 1, 4, 6 (characteristic is defined on p.135), [8], [12] (optional: 7, 18).
    • Section 4.5: 3, 4, 6, [15] (it may help to count elements of order 13), 23, 30, [31] (for A_5 only, plus give one example of a Sylow 2-subgroup for S_5), [32], 35 (optional: 17, 46).
  
  
• Mon Oct 28 -- Fri Nov 1
  • Read sections 5.1, 5.4, 5.5.
  • Worksheet from October 29: Semidirect products.
  • Homework #8 (due Sunday November 17):
    • [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, [10] (note: E_{p^2} is defined on p.155), 14, 15 (optional: 17)
    • Section 5.4: 10, [11], 13
    • Section 5.5: 1, 2, [6], 7bcd, [8] (you may use classification of groups of order p^2, automorphisms from sec 4.4), 18 (optional: 22)
  
  
• Mon Nov 11 -- Fri Nov 15
  • Read sections 5.4, 6.3, 5.2.
  • Worksheet from November 12: 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 18 -- Mon Nov 25
  • Read section 5.2.
  • Worksheet from November 19: Classification of finite abelian groups (continued).
  • Homework #9 (due Friday December 6):
    • Section 5.4: 4, [5], [7], 15, 17 (optional: 9, 14, 19)
    • Section 6.3: 2, 4, [5], 6, [7] (optional: 8)
  • 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
  
  
• Tue Nov 26 -- Fri Nov 29
  • Read sections 7.1, 7.2.
  • Worksheet from November 26: Subrings, ideals, and ring homomorphisms.
  • Homework #10 (due Sunday January 12) -- NOTE: same due date as #11:
    • Section 5.2: 3, [4], [5], 8, [14] (optional: 1, 2, 13, 15)
    • [Problem] Do the analogue of problem 5.2/3 for abelian groups of order 1200 (i.e. find lists of elementary divisors and invariant factors for each such group; how many groups are there up to isomorphism?).
    • Section 7.1: 3, 4, 5, 7, [14], 15 (optional: 9, 21)
  
  
• Mon Dec 2 -- Fri Dec 6
  • Read sections 7.2, 7.3, 7.4, 7.5.
  • We will have a make-up class on Thu Dec 5, 3-5pm at BA1200 (note location!).
  • I will not have regular office hours during exam period.
  • Worksheet from December 3: Fields of fractions.
  • Homework #11 (due Sunday January 12) -- NOTE: same due date as #10:
    • Section 7.1: (optional: 25, 30)
    • Section 7.2: 2, 3, 12 (optional: 6, 7, 8, 13)
    • Section 7.3: 5, 8, [21], [24], 26, 34 (optional: 10, 12, 28, 29)
    • 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) (optional: 11, 14)
    • 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 6 -- Fri Jan 10
  • 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 7: Factorization, GCDs, and Ideals.
  • Homework #12 (due Wednesday January 22):
    • Section 7.4: 19 (use a result from Section 7.1), [25], 30 (note: the first part implies Ex. 7.3/29! why?), [35]
    • 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 13 -- Fri Jan 17
  • 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 14: Factorization in the Gaussian integers.
  • Homework #13 (due Tuesday January 28):
    • Alfonso's notes: 8, 10, [12], [14] (also over the field F₂ of two elements), 15
    • [Problem 1] Show, from the definition of a UFD, 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 (optional: 4)
    • 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 20 -- Fri Jan 24
  • Read Sections 9.1, 9.2, 9.3.
  • Worksheet from January 21: Factorization in polynomial rings.
  • Homework #14 (due Sunday February 2):
    • Section 9.1: [5], 6, 13 (optional: 14)
    • Section 9.2: [1] (note that (f(x)) is not just an ideal but also a subspace, so the quotient is also a vector space), 2, 5, [7] (Hint: what is Z[x]/(2)?), 11 (optional: 3, 7, 10)
    • 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 27 -- Fri Jan 31
  • Read Sections 9.4, 9.5, 13.1 (book), Sections 1-2 of Alfonso's Galois theory notes.
  • Worksheet from January 28: 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 Feb 10, 2020). Please let me know if you find any typos!
  
  
• Mon Feb 3 -- Fri Feb 7
  • Read Sections 13.1, 13.2 (book), Section 1-2 of Alfonso's Galois theory notes (posted just above).
  • Homework #15 (due Sunday February 16):
    • 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 10 -- Fri Feb 14
  • Read Sections 2-3 of Alfonso's notes OR Sections 13.2, 13.3 (book).
  • Worksheet from February 11: Constructions with straightedge and compass.
  
  
• Mon Feb 24 -- Fri Feb 28
  • Read Sections 3-4 of Alfonso's notes OR Sections 13.3, some of 14.1 (book).
  • Worksheet from February 25: Constructions with straightedge and compass. (continued)
  • Homework #16 (due Sunday March 8):
    • Section 13.2: 1, 4, 5 (can you use the tower law?), [7], 8, 9, [10], 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.)
  • For fun: more on straightedge and compass constructions.
  • If you are interested, here are short (but very clever!) proofs that e and pi are transcendental. (See also here for pi.)
  
  
• Mon Mar 2 -- Fri Mar 6
  • Read Sections 4-5 of Alfonso's notes OR Sections 13.4, 14.1 (book).
  • Worksheet from March 3: The Galois group.
  • Homework #17 (due Sunday March 15):
    • Section 14.1: 4, 7, 9 (this is easier if char k is 0), [10]
    • [Problems] in this Handout
  • Here or here is the biography of Galois (1811-1832).
  
  
• Mon Mar 9 -- Fri Mar 13
  • Read Sections 5-6 of Alfonso's notes OR Sections 13.4, 13.5 (book).
  • Worksheet from March 10: Splitting fields and normal extensions.
  • Homework #18 (due Sunday March 22):
    • Section 13.4: 1, 2, [3] (careful when calculating the degree!), 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 16 -- Fri Mar 20
  • 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 17: An example of the FTGT.
  • Homework #19 (due Tuesday March 31):
    • Section 13.5: 1, 7, 11 (optional: 8)
    • 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)).)
    • [Problem] in this Handout
    • 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 23 -- Fri Mar 27
  • 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 24: Finite fields.
  • Homework #20 (due Sunday 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 Mar 30 -- Fri Apr 3
  • Read Sections 10 and 8 of Alfonso's notes OR Sections 14.6/14.7, 14.5 (book).
  • Worksheet from March 31: Computing Galois groups.