AlgebraI

Algebra 1, MATH235 page last update: December 4, 2007.

Instructor: Eyal Goren
Time and location: MWF 8:35-9:25 in the basement of Burnside Building 1B45
Office hours: Monday 13:30 - 14:30, BURN 1108.

TA: Gabriel Chenevert and Telyn Kusalik
Tutorial Hours: Mon 16:30 - 18:00 in BURN 1214 (Telyn), Tue 16:30 - 18:00 in BURN 1214 (Gabriel).
TA office hours: Mon 15:00 - 16:00 (Telyn), Fri 13:00 - 14:00 (Gabriel)
MATH help desk: BURN 911. Schedule is on the door.

Syllabus (Calendar description): Sets, functions and relations. Methods of proof. Complex numbers. Divisibility theory for integers and modular arithmetic. Divisibility theory for polynomials. Rings, ideals and quotient rings. Fields and construction of fields from polynomial rings. Groups, subgroups and cosets; group actions on sets.
Prerequisites: MATH 133 or equivalent
Text book: The course notes, available for free from this website, are the official textbook for this course. The notes will be updated and expanded during the semester. This is the final version of the notes. I shall appreciate you pointing out any typos and inaccuracies. Corrections will be posted here. The course material is everything covered in the notes (but see below regarding the final exam). Some topics will be discussed very briefly in class and you need to study the notes for a more thorough discussion. This will mostly be the case for topics that most of the class have seen before (for example: sets and operations on sets, functions, complex numbers, induction) and will mostly happen in the first month.
Additional text books: You may want to consult additional texts, but be mindful that notation and definitions may be slightly different. Some texts are the following (the library contains many others, not necessarily inferior):

Thomas W. Hungerford: Abstract algebra : an introduction (2nd edition). Thomson Brooks/Cole.
W.E. Deskins:Abstract Algebra.

Evaluation method:

10% Assignments.(I will expect you to do routine easy exercises yourself. The assignments sheets will contain such exercises and you will not need to submit them. Additional easy exercises can be found in Hungerford, for example. The assigenments will contain also more challenging problems and those you will be requested to submit.)
30% Quizzes. There are going to be 3 quizzes during the term. Each will contain multiple choice questions and one proof question. Each quiz is 90 minutes long and will be held in the afternoon or evening. See dates below. The material for Quiz 1 is everything in the table below, until it says (Quiz 1). For Quiz 2, it is the material listed between (Quiz 1) and (Quiz 2), etc.
60% Final exam.
Calculation of final grade: The better of the method described above and the grade of the final exam alone. Same rule applies to deferred/supplemental exams. Based on past experience, I strongly recommend the strategy of getting a high grade on the assignments and quizzes, instead of gambling on just writing a good final. Students will NOT have the option of make-up/additional work to improve their grades.

Academic integrity: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

Solutions and grades of Quizzes
Quiz 1	Results
Quiz 2	Results
Quiz 3	Results

Detailed Syllabus *
Date	Material	Assignment	Misc.
September 5-7	Sets, Methods of Proof.
September 10-14	Methods of Proof. Functions, On the notion of cardinality,	Assignment 1 Solutions	In question 1, all the sets [-n, n] and (n, n+2) and so on stand for intervals of real numbers. In the definition of the A_n you may assume n is a positive integer.
September 17-21	Complex numbers, Polynomials and the fundamental thm of Algebra. Rings and Fields.	Assignment 2 Solutions
September 24-28	Divisibility, gcd, Euclidean algorithm for integers. 2^(1/2) is irrational. Primes and the sieve of Eratosthenes. The Fundamental Thm of Arithmetic.	Assignment 3 Solutions
October 1-5	Infinity of primes (Quiz 1). Equivalence relations. Congruences.	Assignment 4 Solutions	Quiz 1, Wednesday October 3, 16:30 - 18:00 in ENGMD 276 (A-G), ENGMD 279 (H-R), ENGMD 280 (S-Z)
October 10-12	Fermat's little theorem, computing and solving equations in Z_n. Public Key crypto and RSA.The ring of polynomials over a field F. Degree. Division with residue.	Assignment 5 Solutions	No class October 8 (Thanksgiving). Tuesday follows Monday's schedule. A correction was made in question 4
October 15-19	GCD's. The Euclidean algorithm for polynomials. Irreducible polynomials. Unique factorization. Roots of polynomials. Roots of rational and real polynomials. Roots of polynomials over Z_p.	Assignment 6 Solutions	Note that you have 2 weeks to submit this (somewhat long) assignment.
October 22-26	Rings (recall). Ideals. Z and F[x] are principal ideal rings. (Quiz 2). Homomorphisms and kernels.		Quiz 2, Friday October 26, 16:30 - 18:00 in STBIO S1/4. Material: sections 11 - 17 in the course note (watch for an updated version over the weekend). Structure: similar but 2 proofs this time, worth 44%. At this point, the material in the course gets much more abstract and sophisticated. I very much recommend to all those skipping classes to join the class again.
October 29 - November 2	Quotient rings. F[x]/(f(x)) and constructing finite fields. Roots in extension fields.	Assignment 7 Solutions
November 5-9	First isomorphism theorem. Chinese remainder theorem. Applications of CRT. Groups: the basic definition and examples. The symmetric group. (Quiz 3).	Assignment 8 Solutions	On this week (only!) it is possible to review your quizzes during the office hours of Gabriel, Telyn or Eyal. Questions 2, 3 in assignment 8 are optional. (They are not hard. I am making them optional, i.e., for extra credit, just to make the assignment a little shorter.)
November 12-16	The symmetric group - cont'd. (Quiz 3).The dihedral group. Cosets and Lagrange's theorem.	Assignment 9 Solutions	Note that there is now a new version of the solutions to ass. 9, correcting some errors. Quiz 3, Wednesday November 14, 16:30 - 18:00 in STBIO S1/4. Course evaluations are now open. I will appreciate you taking the time to complete the evaluations. Your comments, positive and negative, will be taken very seriously and will help me realize what's working and what's not. It is your chance to provide criticism and praise.
November 19-23	Homomorphisms and isomorphisms; Cayley's theorem. Group actions on sets: first definitions and properties. Examples.	Assignment 10 Solutions	This is the last assignment!! Question 5 is optional.
November 26-30	Cauchy-Frobenius formula. Applications to Combinatorics. Homomorphisms, normal subgroups, quotient groups and the first isomorphism theorem. Examples.
December 3	Groups of low order.

How to prepare for the final exam?
The structure and nature of the final exam is similar to the quizzes and is of similar level of difficulty, except that the proofs you'll be asked to reproduce are harder than those you had to do in the quizzes, I think.
Specifically:

Part I consists of 12 multiple choice questions, each worth 4 points for a maximum of 44 points. Each question has a unique correct answer. Penalty of -2 for a wrong answer (so if you can eliminate half or more of the possible answers, statistically it is a good strategy to guess).

Part II consists of 3 questions, each with two or more parts (depends on the question). Each question requires you to give a proof of a Theorem/Proposition/Lemma that we proved in class and, in addition, to perform some related computations.

The final covers all the material done in the class, the course notes (excluding "planned sections" and Cauchy's theorem (section 28) and including section 29, but only to the extent we have covered it in class). The final also covers all the assignments, including the last assignment!
Although, in principle, the final can touch any topic, there is certainly emphasis of the material in sections 12 -27 (inclusive). Very heavy emphasis on groups, constructing fields and working in them, and congruences. Though, again, this does not mean that other topics are not covered in the final!

Special Office hours prior to the exam:

Day (all in the week Dec. 3 -7). No office hours Mon. Dec. 10	Time
Monday	1:30-2:30 (Eyal), 3:00 - 4:00 (Telyn)
Tuesday	1:00 - 2:00 (Gabriel), 2:00 - 3:00 (Eyal)
Wednesday	1:00 - 2:00 (Telyn)
Thursday	9:00 - 10:00 (Eyal), 1:00 - 2:00 (Telyn)
Friday	11:00 - 12:00 (Gabriel), 1:00 - 2:00 (Gabriel)