Schedule
This is a tentative schedule for the course. If necessary, it may change.
 01/09/12 Chap 12: Definition of ring, examples. Here is a summary.
 01/11/12 Chap 12,13: Basic properties of rings. Subrings. Here is a summary.
 01/13/12 Chap 13: Integral domains, division rings and Fields. Here is a summary. There are extra examples that you need to learn.
 01/16/12 NO CLASS
 01/18/12 Chap 14: Integral domains, characteristic. Here is a summary.
 01/20/12 Chap 14,15: Ideals, homomorphisms, kernel and image, the characteristic homomorphism Z to R for a unital ring R. Here is a summary.
 01/23/12 Chap 14,15: Isomorphisms, Factor rings, Ideals = kernels. Here is a summary.
 01/25/12 Chap 15: 1st homomorphism theorem, ideals generated by a subset, homomorphic images of Z. Here is a summary.
 01/27/12 Chap 14,15: PIR and PID. Z[x] is not a PID. Here is a summary.
 01/30/12 Chap 14,15: Prime and maximal ideals. Here is a summary.
 02/01/12 Exam I: See the midterm section for more information. Here are solutions to most of the problems.
 02/03/12 Chap 15: Review of the exam. Field of quotients of an integral domain.
 02/06/12 Chap 16: Field of quotients of an integral domain. Here is a summary.
 02/08/12 Chap 16: Polynomial rings. Division algorithm for polynomials. F[x] is PID. Here is a summary.
 02/10/12 Chap 16: More on polynomials. Remainder theorem.
 02/13/12 Chap 17: Factor theorem. A polynomial of degree n has at most n roots. Here is a summary.
 02/15/12 Chap 17: Evaluation. R[x]/(x^{2} + 1) is isomorphic to C. Irreducible polynomials. Principal ideals (f) are maximal if and only if f is irreducible. Here is a summary.
 02/17/12 Chap 18: Irreducibility of deg 2 and 3. Gauss's lemma. Here is a summary.
 02/20/12 NO CLASS
 02/22/12 Chap 18: Irreducibility Z and Q. Irreducibility mod p and Q. Here is a summary.
 02/24/12 Chap 18: Irreducibility mod p and Q (continue). Irreducibility and prime elements in rings. Here is a summary.
 02/27/12 Chap 18: Irreducibility and prime elements in rings (continue). Here is a summary.
 02/29/12 Exam Review. Here is a summary.
 03/02/12 Exam II: see the midterm section for a practice exam. (There are two versions of this exam. Here is the second version.)
 03/05/12 Chap 18: A PID is a UFD. Here is a summary.
 03/07/12 Chap 1920: A PID is a UFD (continue). Rings of the form Z[sqrt(d)] and Pell's equation. Euclidean domains. Here is a summary.
 03/09/12 Chap 2021: A ED is a PID (continue). Irreducibles in Z[sqrt{d}]. Z[sqrt{10}] is not UFD. Here is a summary. Some corrections are made.
 03/12/12 Chap 21: Z[i] is ED. Vector Spaces. Here is a summary.
 03/14/12 Chap 21: Field extensions. Splitting Fields over Q. Big Theorem on Field extensions F(α).
 03/16/12 Review day
 03/17/12 Review day (9:3010:30)
 03/19/12 FINAL EXAM

Assignments.
 Due 01/13:
 Chapter 12: Problems 2, 6, 20, 22, 26, 32, 40, 46.
 Let p, p_{1}, ... , p_{n} be distinct primes.
 Prove that any unital ring of order p^{2} is commutative.
 Give an example of a noncommutative ring of order p^{2}.
 Prove that any ring of order p_{1}p_{2}...p_{n} is commutative.

Give an example of a ring R with an element r such that r is not invertible and at same time it has a left inverse.

Think about these problems. But they are NOT part of the problem set: Chapter 12: Problems 1, 3, 23, 36, 45, 49, 51.
 Due 01/20:
 Chapter 12: 18, 24, 48, 50.
 Chapter 13: 4, 8, 14, 16, 26.
 Prove that the quaternion ring given in the class is a division ring.
 Find U(Z[i]).
 Prove that a finite integral domain is a field.

Think about these problems. But they are NOT part of the problem set:
 Chapter 12: Problems 17, 19, 37, 39, 51.
 Chapter 13: Problems 5, 7, 9.
 Due 01/27:
 Chapter 13: 20, 24, 30, 40, 45, 47, 58.
 Chapter 14: 4, 8, 10, 12, 14.
 Let R be ring. We define the center Z(R) as {x∈ R for any y∈R, xy=yx}.
 Prove that Z(R) is a subring.Give an example where Z(R) is not an ideal.
 Prove that, if D is a division ring, then Z(D) is a field.

Let R be a unital ring. Prove that J is an ideal of M_{n}(R) if and only if
there is an ideal I of R such that J=M_{n}(I). In particular, if D is a division ring,
then the only proper ideal of M_{n}(D) is the trivial ideal.

Think about these problems. But they are NOT part of the problem set:
 Chapter 13: Problems 33, 35, 37.
 Chapter 14: Problem 7, read Example 11 in Page 265.
 Due 02/10:
 Part 6 of Problem 2 in the exam.
 Chapter 14: 24, 32, 34, 38, 62
 Supplementary exercises for Chapters 1214: 42, 48
 Chapter 15: 42, 46, 66, 68
 Due 02/17:
 Chapter 15: 56.
 Chapter 16: 2, 12, 33, 34, 42, 47, 50, 51
 Due 02/24:
 Chapter 16: 32, 41.
 Chapter 17: 6, 8, 10, 25, 31, 32
 Chapter 17: 15, 16, 17 (You need each problem for the next one! At the end you prove a nice fact!)
 Due 03/9:
 Due 03/16:
 In this problem, you will prove that Z[x] is a UFD.

Prove if either f(x) is a prime in Z or f(x) is primitive and irreducible over Q, then f(x) is irreducible in Z[x].

Prove if deg(f)>0 and f is irreducible in Z[x], then f is primitive and irreducible over Q.

Prove if deg(f)=0 and f is irreducible in Z[x], then f is a prime in Z.
 Remark: the above three steps, show that
f(x) is irreducible in Z[x] if and only if either f(x) is a prime in Z or f(x) is primitive
and irreducible over Q.
 Prove if f,g in Z[x] are primitive and f=cg for some rational number c, then f=± g.
 Using the above step conclude that if f,g in Z[x] are primitive and f and g are associates in Q[x],
then f and g are associates in Z[x].
 In class using Gauss's lemma we essentially proved that if f is primitive, g_{1}, g_{2} in Q[x] and f(x)=g_{1}(x)g_{2}(x), then there is a rational number c such that
 cg_{1}(x) and (1/c)g_{2}(x) are primitive polynomials.
In what follows you are allowed to use this statement.
 If f is primitive, g_{i}'s are in Q[x] and f(x)=g_{1}(x)g_{2}(x)...g_{n}(x), then there are rational numbers c_{i} such that
 For any i, c_{i}g_{i}(x) is primitive.
 c_{1}.c_{2}...c_{n}=1
 Use the above steps and the facts that Z and Q[x] are UFDs, to prove that Z[x] is a UFD.
 In this problem you will prove that Z[i] is a Euclidean Domain. You will show that d:Z[i]→ Z^{≥0}
given as d(x+iy)=x^{2}+y^{2} satisfies the needed properties.
 Prove d(zz')≥ d(z) for any z,z' in Z[i].
 Let z=x+iy and z'=x'+iy' be in Z[i] and assume that z' is not zero. Then prove that there are rational numbers s and t such that
z=z'(s+it). (Hint: Q[i] is a field.)

For any rational number s there is an integer n such that sn is at most 1/2.

For any z,z' in Z[i], if z' is not zero, then there are z'' in Z[i], r_{1} and r_{2} in Q such that
 z=z'(z''+ (r_{1}+i r_{2})).
 r_{1} and r_{2} are at most 1/2.
 Let r=z'(r_{1}+i r_{2}). Then r is in Z[i] and d(r)< d(z').
 Conclude that for any z,z' in Z[i], if z' is not zero, then there are q and r in Z[i] such that
which implies that Z[i] is a ED.
 Chapter 18: Problems 14, 18, 22.
 Supplementary Exercises for Chapters 1518: Problems 3, 4, 11, 30.
