Math 401

Spring 2015

A first course in rings and groups out of Shifrin.

• Syllabus.
• Errata for the textbook.
• Handout on Writing by Jack Lee.
• Reading 1 (due 1/11): Section 1.2.
• Homework 1 (due 1/13):
  With a partner and using a computer, find expressions for the following as polynomials in n:
  • 1⁵ + 2⁵ + … + n⁵
  • 1⁶ + 2⁶ + … + n⁶
  • However many more you feel like doing.
  Comment on any patterns that you observe. Prove one of your expressions by induction.
• Reading 2 (due 1/18): Section 1.3.
• Homework 2 (due 1/22):
  • Section 1.2 #1 (choose three parts), 3, 7, 12, 13.
  • Section 3.1 #2 (choose two parts). The Euclidean algorithm works for polynomials the same as it does for numbers, so this is just like 1.2 #1.
• Homework 3 (due 1/29):
  • Section 1.3 #5, 14, 20 (choose three parts).
• Homework 3½ (due 2/3): Do one or both:
  • Compose a message, encode it in blocks of nine letters using Shifrin's code A = 01, B = 02, …, space = 00, period = 27, encrypt it using RSA with N = 340196134436655019 and e = 3, and send it to me. I will send you back the plain text.
  • Choose prime numbers p and q such that k = (p-1)(q-1) is relatively prime to e = 3, and compute d. Send me N = pq. (Do not send me p, q, k, or d!) I will send you a secret message to decrypt and decode. Bonus points if my computer can't factor your N in less than a minute. (But no bonus points if I can google your number.)
• Homework 4 (due 2/5):
  • Let R be a commutative ring and a, b, d ∈ R. Show that if (a,b) = (d) then d is a greatest common divisor of a and b.
  • Let R be an integral domain and a, b ∈ R. Show that (a) = (b) if and only if there is a unit u in R such that a = ub.
• Reading 3 (due 2/15): Sections 3.3 and 4.1.
• Homework 5 (due 2/19):
  • Section 1.3 #21 (choose three parts)
  • Section 1.4 #4
  • Section 2.2 #11a
  • Section 2.3 #6 and 9 (choose two parts)
  • Section 2.4 #6 (choose one part; ignore the thing about Δ and Proposition 4.4)
  • Section 3.1 #10
  • Seciton 3.2 #10
• First midterm has been moved to Thursday 2/19 Tuesday 2/24.
• Midterm 1, Solutions, tex.
• Reading 4 (due 3/15): Section 4.3.
• Homework 6 (due 3/19):
  • Section 3.3 #2 (do half, and use the table on page 112 if you want), 9 (optional), 10.
  • Section 4.1 #8 and 17. To #17 add the following: Let K ⊂ R be a third ideal; show that if I ⊂ K and J ⊂ K then I + J ⊂ K; observe that if K ⊂ I and K ⊂ J then K ⊂ I ∩ J (this is trivial); interpret these in terms of “a divides b” type statements. Show that IJ ⊂ I ∩ J, but give an example where IJ ≠ I ∩ J. Find the sum and intersection of each pair of our three favorite ideals in Z[√-5]: I = (2, 1+√-5) = (2, 1-√-5) = (1+√-5, 1-√-5), J₁ = (3, 1+√-5), and J₂ = (3, 1-√-5).
  • Section 4.2 #1, 4b, 5, 8.
• Reading 5 (due 3/29): Section 6.1.
• Homework 7 (due 4/2):
  • Section 4.2 #21.
  • Section 4.3 #1 (choose one part), 14 (this is not a typo), 2, 8, 15.
  • Section 5.2 #7, 13 (hint: 72° = 2π/5 is constructible by exercise 2.4.10).
• Second midterm has been moved to Tuesday 4/7 Thursday 4/9.
• Midterm 2, Solutions, tex.
• Homework 7 (due 4/27):
  1. Let's understand the group of units in the ring Z_n for n ≤ 20. The Chinese Remainder Theorem says that if m and n are relatively prime then Z_m × Z_n is isomorphic to Z_mn as rings. Thus in particular the additive group Z_m × Z_n is isomorphic to Z_mn, and the group of units Z_m^× × Z_n^× is isomorphic to Z_mn^×.
     1. Show that if R and S are rings then the group of units (R × S)^× is isomorphic to R^× × S^×.
     2. Find the orders of Z₂^×, Z₄^×, Z₈^×, and Z₁₆^×. Is Z₈^× isomorphic to the additive group Z₄, or to Z₂ × Z₂? Is Z₁₆^× isomorphic to Z₈, or to Z₄ × Z₂, or to Z₂ × Z₂ × Z₂? Why are those last three not isomorphic to one another?
     3. Find the orders of Z₃^× and Z₉^×. What additive groups are they isomorphic to?
     4. For all n ≤ 20, find the order of Z_n^×, and decide which additive group it is isomorphic to.
  2. Let G be a group. Show that the map φ: G → G defined by φ(g) = g² is a homomorphism if and only if G is abelian.
  3. Let G be a group and H be a subgroup. Show that the following are equivalent:
     1. For all g ∈ G we have gHg^-1 = H.
     2. For all g ∈ G we have gHg^-1 ⊂ H.
  4. Let D₅ act on itself by conjugation. What are the orbits? Pick one element of each orbit and find its stabilizer. Use the notation
     D₅ = { 1, r, r², r³, r⁴, s, sr, sr², sr³, sr⁴ }
     and the fact that r^ks = sr^-k. Do the same for D₆ and S₅. With S₅ use cycle notation.
  • Finally, do section 6.1 #22 and 25 and section 6.3 #25.
• Office hours Thursday 4/23, 1:00–3:00 and Friday 4/24 10:00–11:30.
• Final Exam, Solutions, tex.