Math 71

Algebra

Instructor: Carl Pomerance (carl.pomerance@dartmouth.edu)

### News and current assignment

Please hand in your test at my office or leave in my mailbox.

On problem 4 on the test, you are permitted to assume that not only is n composite, but that it is divisible by at least two different primes. You will get 5 points extra credit if you solve it as originally stated.

### Abstract

The theme of the course is abstract algebra. Most people think of algebra as that part of math where you solve equations or manipulate expressions which have letters in them. This actually is not all that closely related to abstract algebra, though there is a connection. Abstract algebra deals with the structures of familiar objects, such as the integers, the complex numbers, matrices, etc. The idea is that when these more concrete sets are studied in the abstract, certain of their properties are forced upon them because of other properties already assumed. Thus, one can then deduce these other properties for any system that has the basic assumed properties. Depending on how many of these basic axioms are assumed, we have groups, rings, and fields. These will be our object of study.

A very brief syllabus (chapters from Dummit and Foote, Abstract Algebra):
In week 1, we will study Chapter 0 (equivalence relations, integers).
In the following 4 weeks, we will basically do a chapter per week, with a few sections skipped.
a small amount of time on fields.

### Classes

Room: 108 Kemeny
Lectures: Monday-Wednesday-Friday 10:00 am--11:05 am (10 hour)
X-period: Thursday 12:00 noon--12:50 pm
There are tutorials held 7 PM to 8 PM Sundays in 343 Kemeny.

### Staff

Instructor:
Carl Pomerance -- 339 Kemeny Hall
Office hours: Tuesday, Wednesday, Thursday 9:00am--10:00am and by arrangement at other times.
Yangyang Liu

### Textbook

Abstract Algebra, 3rd edition, by Dummit and Foote
(required)
See a short list of errata at http://www.emba.uvm.edu/~foote/errata_3rd_edition.pdf

This book is available at Wheelock Books and elsewhere. (Yes, I know its expensive.)

Homework 20%, two mid term exams each 20%, final exam 40%. As much as possible, grades will be based on demonstrated knowledge. However relative performance may be used as a criterion for increasing grades, and grade borderlines will be chosen to place a relatively small number of students on borderlines. At the end of the term, the lowest of your 4 grades (hw, midterms, final) will be dropped, except if your final exam is your lowest grade, in which case the weight of the final exam will be halved. (So, if one of the midterms or hw is dropped, then the remaining 3 grades have weights 1/4, 1/4, 1/2; while if the final is the lowest, the four grades have equal weight 1/4, 1/4, 1/4, 1/4.)

### Homework

Homework is due at the start of the class period on the due date. Late homework is generally not accepted unless there is a prior arrangement.

### Past assignments

Homework due Monday, Sept. 25:
Section 0.1, #1-5 (Try to do 2 and 3 in a "coordinate-free" way. You are allowed to assume that matrix multiplication is associative and that it satisfies the distributive law with matrix addition.)
Section 0.2, #2, 5
Section 0.3, #5-7

Homework due Monday, Oct. 2:
Section 1.1, #1b, 1d, 2b, 2d, 9, 18, 19, 22, 25.
Extra problem: Show that if G is a group and x is an element of G of finite order n, then for all integers a, b, we have xa = xb if and only if a &equiv b (mod n).

Section 1.2, #1, 3; Section 1.3, #1, 3, 15.
Instead of the hint for number 15, which I didn't understand, prove that if A, B are disjoint subsets of {1,2,...,n} and if &sigma and &tau are in Sn with &sigma moving only members of A and &tau moving only members of B, then &sigma&tau=&tau&sigma and |&sigma&tau| = lcm[|&sigma|,|&tau|].

Homework due October 9:
Section 1.4, #4, 5, 8
Section 1.5, #1
Section 1.6, #3-8, 17-20
Show that if H is a subgroup of the finite group G that is not G itself, then |H| &le |G|/2.
(Hint: Let x be an element of G that is not in H, and show that no element hx can be in H as h runs over the elements of |H|.)

Homework due Wednesday, Oct. 18:
Section 3.1, #1, 3 -8, 14. (The roots of unity in the complex numbers are those complex numbers z for which zn=1 for some positive integer n. They form a group under multiplication.)
Sec. 2.2, #3, 9.
Sec. 3.2, #5, 6.

Homework due Monday, Oct. 23:
Sec. 3.3, #1, 3, 7, 8.
Sec. 3.5, #2, 3, 4, 5, 12, 13.

Homework due Monday, October 30:
Sec. 4.1, #1, 2, 3.
Sec. 4.2, #2, 8, 9.
Concerning 4.2.2, the "left regular representation of a group G of order n into Sn" is the same thing we did when we proved Cayley's theorem. Namely, each element g of G acts on G by left multiplication and so can be associated with a particular permutation on n letters.

Homework assignment due Monday, Nov. 13:
Section 4.5, #1-3, 18-23.
Section 7.1, #1-5, 15 [Hint: Show each element is its own additive inverse.]

Homework due Monday, Nov. 20:
Section 7.2, #1
Section 7.3, #1-4, 8, 10
Section 7.4, #4, 8, 10 (assume R has 1 and at least 2 elements)
Section 8.1, #3, 4a

### Exams

The two midterm exams will be open-book, open-notes take-home tests and will be given out on October 11 and November 1, each due in class two days later.

The final exam will also be open-book, open-notes take-home. It will be given out on the last day of classes and will be due by 11:00 AM on December 4 (the end of our regularly scheduled exam period).

### Honor Code

Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. Students are discouraged from using solutions to problems that may be posted on the web, and as just stated, must reference them if they use them. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's code or solutions, in whole or in part, is a violation of the Honor Code.

The honor principle applies to exams as follows: Students may not give or receive assistance of any kind on an exam from any person except the professor or someone explicitly designated by the professor to answer questions about the exam. Students may not use a computer during an exam, but they may use a calculator to help with simple arithmetic.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me, and I will be glad to help clarify things. It is always easier to ask beforehand.

### Other

I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.

I realize that some students may wish to take part in religious observances that fall during this academic term. Should you have a religious observance that conflicts with your participation in the course, please come speak with me before the end of the second week of the term to discuss appropriate accommodations.

This page was inspired by the web site for Math 19 in Fall 04, written by Alin Popescu