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*:- 1
^{5}+ 2^{5}+ … +*n*^{5} - 1
^{6}+ 2^{6}+ … +*n*^{6} - However many more you feel like doing.

- 1
- 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.)

- 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
- 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*.

- Let
- 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),*J*_{1}= (3, 1+√-5), and*J*_{2}= (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):
- 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}^{×}.- Show that if
*R*and*S*are rings then the group of units (*R*×*S*)^{×}is isomorphic to*R*^{×}×*S*^{×}. - Find the orders of
**Z**_{2}^{×},**Z**_{4}^{×},**Z**_{8}^{×}, and**Z**_{16}^{×}. Is**Z**_{8}^{×}isomorphic to the additive group**Z**_{4}, or to**Z**_{2}×**Z**_{2}? Is**Z**_{16}^{×}isomorphic to**Z**_{8}, or to**Z**_{4}×**Z**_{2}, or to**Z**_{2}×**Z**_{2}×**Z**_{2}? Why are those last three not isomorphic to one another? - Find the orders of
**Z**_{3}^{×}and**Z**_{9}^{×}. What additive groups are they isomorphic to? - For all
*n*≤ 20, find the order of**Z**_{n}^{×}, and decide which additive group it is isomorphic to.

- Show that if
- Let
*G*be a group. Show that the map φ:*G*→*G*defined by φ(*g*) =*g*^{2}is a homomorphism if and only if*G*is abelian. - Let
*G*be a group and*H*be a subgroup. Show that the following are equivalent:- For all
*g*∈*G*we have*gHg*^{-1}=*H*. - For all
*g*∈*G*we have*gHg*^{-1}⊂*H*.

- For all
- Let
*D*_{5}act on itself by conjugation. What are the orbits? Pick one element of each orbit and find its stabilizer. Use the notation

*D*_{5}= { 1,*r*,*r*^{2},*r*^{3},*r*^{4},*s*,*sr*,*sr*^{2},*sr*^{3},*sr*^{4}}

and the fact that*r*=^{k}s*sr*. Do the same for^{-k}*D*_{6}and*S*_{5}. With*S*_{5}use cycle notation.

- Finally, do section 6.1 #22 and 25 and section 6.3 #25.

- Let's understand the group of units in the ring
- Office hours Thursday 4/23, 1:00–3:00 and Friday 4/24 10:00–11:30.
- Final Exam, Solutions, tex.