# 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, dR. 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, bR. 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 KR be a third ideal; show that if IK and JK then I + JK; observe that if KI and KJ then KIJ (this is trivial); interpret these in terms of “a divides b” type statements. Show that IJIJ, but give an example where IJIJ. 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 Zn for n ≤ 20. The Chinese Remainder Theorem says that if m and n are relatively prime then Zm × Zn is isomorphic to Zmn as rings. Thus in particular the additive group Zm × Zn is isomorphic to Zmn, and the group of units Zm× × Zn× is isomorphic to Zmn×.
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 Zn×, and decide which additive group it is isomorphic to.
2. Let G be a group. Show that the map φ: GG 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 gG we have gHg-1 = H.
2. For all gG we have gHg-1H.
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 rks = 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.