**
**Math 103B (Driver, Spring 2009) Applied Modern Algebra

*
(http://math.ucsd.edu/~bdriver/103B_S09/index.htm)*

**
**The final exam is on Thursday June 11 from 3pm-6pm
in
HSS 1305.

This final is cumulative.

Please bring a blue book!

You may bring one 8 1/2 X 11 sheet (front and back) of notes if you wish.

B. Driver's finals Week Office Hours
are:

M., Tu. & W. at 11:00 AM -- 12:00 Noon.

J. Reed's finals Week Office Hours
are:

Monday
4:30-5:30,Tuesday 1:00-2:00, and Wednesday 1:00-2:00.

**Instructor: **Bruce
Driver (bdriver@math.ucsd.edu),
APM 7414, 534-2648.** Office Hours: ****M & W
5:00-6:00PM and F 2:30-3:30 PM.**

**TA: **Joseph Reed (j2reed@math.ucsd.edu),
APM 5801, 534-9054 . **Office Hours: ****
Tuesday, 12-1.**

**Meeting times: **MWF 4:00p - 4:50p in
HSS 1305 . DI (A01) Th 5:00p - 5:50p CENTR 201.

**Textbook:** *Contemporary Abstract
Algebra* by Gallian, 6th Edition.

**Prerequisites:** The prerequisites are Math 20F, Math 109, and Math 103A
(or 100A).

**Homework:** Homework will be assigned
weekly; the list of problems for the week will be posted on the class website.
Homework will be due in the Thursday sections. Late homework will not be accepted, but
the lowest homework score will be dropped.

**Exams: **There will be weekly 20 minute quizzes at the
**start of each Friday class **(excluding the first Friday, April 3.) The final exam
is on
**Thursday June 11** from 3pm-6pm in
HSS 1305. No books, class notes, or calculators are allowed during
the quizzes or final exam. The Final Exam will be **cumulative.**

**Grading: **
*
***Final Grade **= homework
(25%) + Quizzes (40% for all) + final (35%).

**Description: **Math 103B is the second quarter of a two quarter
sequence. While the material from Math 103A will be referred to only
occasionally, if you have not taken that course in a while you may need to
review the basic facts about groups. The main topic of 103B is the theory of
rings and fields, and as an application, algebraic coding theory. We will cover
roughly chapters 12-23 and chapter 31 of Gallian's text.

**Academic honesty:** I expect you to abide by the university's policies.
Serious cases of dishonesty may be reported to the appropriate university
committee. The most straightforward kind of cheating which is obviously
disallowed is copying from a neighbor's exam, or consulting notes or the book
during an exam.

The honesty rules for homework are sometimes less obvious. So there is no
confusion, here are my particular rules.

- The homework you hand in should be your own written work, and your own
only. It is not acceptable to copy word for word, or paraphrase, the work of
another student in the class, or a solution found (say) on the internet or
in a solutions manual, and hand it in as your own work.

- You may work with others in the class (I am all for this), but be
careful not to violate rule 1 above. Certainly you can freely discuss
definitions, examples in the text, etc. with others to help you understand
them. For the homework problems, it is best to start by thinking about the
problems yourself, hard. You may see how to do some of them, but be stuck on
others. Wait a while, you will probably have additional insights the next
day (this is one reason it is important to start the homework early). You
can ask us or a friend for hints. Hopefully after more thought you will see
how to solve the ones you were stuck on. If not, and here and there a friend
tells you how to do a problem you were completely stuck on, and then you
write it up yourself using only your own understanding, that's OK. But this
should only be a problem here or there, not a significant fraction of them,
or else you won't learn how to work through these problems independently.

- Just to clarify further, reading through a friend's entire solution to a
problem which you did not think about yourself and then immediately writing
your own solution is not allowed. You are likely to end up writing a
paraphrase of the other solution and not really understand the proof, and
you won't gain the benefit that comes from thinking long and hard about the
problems.

**Tentative Topics List: **The following outline of what we will cover
and the order is subject to change.

Chap 12: Definition of ring, examples.

Chap 12,13: Basic properties of rings. Subrings. (Integral) domains.

Chap 13: Fields. Characteristic.

Chap 14: Ideals, factor rings.

Chap 14: Examples of factor rings. Prime and maximal ideals.

Chap 15: Homomorphisms. 1st homomorphism theorem.

Chap 15: The characteristic homomorphism Z to R for a ring R. Ideals = kernels.

Chap 15: Field of quotients of a commutative domain.

Chap 16: Introduction to polynomial rings. + Exam Review.

Chap 16: Division algorithm for polynomials.

Chap 16: F[x] is a PID. Evaluation. R[x]/(x^2 + 1) is isomorphic to C.

Chap 16,17: Remainder and Factor theorems. A polynomial of degree n has at most
n roots. Irreducible polynomials.

Chap 17: Irreducibility tests.

Chap 17: Proof of mod p test. Principal ideals (f) are maximal if and only if f
is irreducible. Construction of finite fields.

Chap 18: Irreducibility and prime elements in rings, UFDs.

Chap 18: Rings of the form Z[sqrt(d)].

Chap 18: A PID is a UFD. Euclidean domains.

Chap 18: Z[i] is a Euclidean domain. Z[x] is not a PID.

Chap 19: Vector Spaces

Chap 20-21: Field extensions. Splitting Fields over Q.

Chap 20-21: Big Theorem on Field extensions F(alpha).

Chap 23: Compass and straightedge constructions~~.~~

Chap 31: Error correcting codes part I

Chap 31: Error correcting codes part II