Modern Algebra II

Winter 2012

 Lectures: M-W-F 11:00 AM--11:50 AM SOLIS 109 Office Hour: M-W 1:00 PM--2:00 PM APM 7230 Discussion Session: Tu 1:00 PM--1:50 PM APM B412 TA: James Berglund (jberglunmath ucsd edu)

Book

• J. A. Gallian, Contemporary abstract algebra. (The main textbook)
• P. B. Bhattacharya, S. K. Jain, S. R. Nagpaul, Basic abstract algebra.
• D. Hungerford, Algebra.

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]/(x2 + 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 19-20: 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 20-21: 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 (9:30-10:30)
• 03/19/12 FINAL EXAM

Assignments.

• Due 01/13:
• Chapter 12: Problems 2, 6, 20, 22, 26, 32, 40, 46.
• Let p, p1, ... , pn be distinct primes.
• Prove that any unital ring of order p2 is commutative.
• Give an example of a non-commutative ring of order p2.
• Prove that any ring of order p1p2...pn 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 Mn(R) if and only if there is an ideal I of R such that J=Mn(I). In particular, if D is a division ring, then the only proper ideal of Mn(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 12-14: 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, g1, g2 in Q[x] and f(x)=g1(x)g2(x), then there is a rational number c such that
• cg1(x) and (1/c)g2(x) are primitive polynomials.
In what follows you are allowed to use this statement.
• If f is primitive, gi's are in Q[x] and f(x)=g1(x)g2(x)...gn(x), then there are rational numbers ci such that
• For any i, cigi(x) is primitive.
• c1.c2...cn=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)=x2+y2 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 |s-n| is at most 1/2.
• For any z,z' in Z[i], if z' is not zero, then there are z'' in Z[i], r1 and r2 in Q such that
• z=z'(z''+ (r1+i r2)).
• |r1| and |r2| are at most 1/2.
• Let r=z'(r1+i r2). 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
• z=z'q+r
• d(r)< d(z')
which implies that Z[i] is a ED.
• Chapter 18: Problems 14, 18, 22.
• Supplementary Exercises for Chapters 15-18: Problems 3, 4, 11, 30.

• Your letter grade will be based on the best of the following:
• Homework 10%+ First Midterm 25%+ Second Midterm 25%+ Final 40%
• Homework 10%+ Best of midterms 25%+ Final 65%
• Homework and midterm exams will be returned in the discussion sections.
• If you wish to have your homework or exam regraded, you must return it immediately to your TA.
• Regrade requests will not be considered once the homework or exam leaves the room.
• If you do not retrieve your homework or exam during discussion section, you must arrange to pick it up from your TA within one week after it was returned in order for any regrade request to be considered.
• A good portion of the exams will be based on the weekly problem sets. So it is extremely important for you to make sure that you understand each one of them.
• You are encouraged to work together on the problem sets. You can also go to the office hours and the discussion session to get help with some of the problems. But before that you have to think about them on your own at least for an hour and at the end you have to write down your own solution.
• The worst homework grade will be dropped.
• I might ask you to come to my office and explain your solution to some of the problems.
• Late HW is not acceptable!
Midterm.

• The exam is till the end of Chapter 15, excluding prime and maximal ideals and the field of quotients.
• You have to learn all the examples mentioned in class and in the book.
• You have to go over all the problems in your weekly homework.
• Here is a practice exam.
• Please bring a blue-book for the midterm.
• Here is the first midterm and here are solutions to most of the problems.
• The exam is from Chapter 14 till end of Chapter 17.
• You have to learn all the examples mentioned in class and in the book.
• You have to go over all the problems in your weekly homework.
• Here is a practice exam.
• Please bring a blue-book for the midterm.
• There are two versions of this exam: The first version of Exam II and The second version of Exam II.
• If you want to get A, you should study all the examples mentioned in class and in the book and go through the lecture notes.
• If you are short in time and only want to study some parts of the discussed topics (not a good idea!), then I recommend at least reading the following topics in details
• Irreducibility tests.
• Connections between irreducibles, maximal ideals and constructing fields.
• Solving a polynomial in a larger field.
• Irreducibles, primes and associates and their relations with principal ideals.
• ED implies PID. PID implies UFD.
• How to determine if a given element in Z[\sqrt{d}] is a unit or an irreducible.
• Problems will be similar to the relevant problems in the book and examples mentioned either in class or in the book.

Acknowledgment. I would like to thank Prof. D. Rogalski for sharing his syllabus and views on this course with me. The regrading policy is copied from Prof. A. Ioana's webpage and I am thankful for that.