AlgebraI

Algebra 1, MATH235 page last update: November 30, 2008.

Instructor: Eyal Goren
Office: BURN 1108.
Office hours: Mon 14:00 - 15:00, Wed 15:00 - 16:00.
Time and location of course: MWF 8:35-9:25 in the basement of Burnside Building 1B45

TA's: Victoria de Quehen, Michael Musty
Tutorial Hours: M 16:35 - 17:55, BURN 1214 (MM), T 16:35 - 17:55, BURN 1214 (VdQ)
TA office hours: Th 16:00 - 18:00 (Room 1018) (VdQ),
T - Th 9am-10am (Room 1033) (MM).
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. The course material is everything covered in the notes, except for certain sections to be indicated prior to the 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, which can be found in the notes, or in any textbook in basic algebra. The assignments sheets will contain more challenging problems. Assignments will be posted on Mondays (at the latest) and should be submitted on Mondays before 16:00.
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

PLEASE NOTE: In the assignments, exercise numbers and page numbers refer to the version of the notes at the time the assignment is posted. Since the notes are updated and expanded weekly, you should be careful to look at the current version and not at an older version of the notes you may have. Refresh your browser.

Detailed Syllabus *
Date	Material	Assignment	Misc.
September 3-5	Sets, Methods of Proof. Functions
September 8-12	Functions. On the notion of cardinality. Complex numbers. Polynomials and the fundamental thm of Algebra. Rings and Fields (definition only).	Assignment 1 Solutions
September 15-19	Divisibility, gcd, Euclidean algorithm for integers. 2^(1/2) is irrational. Infinity of primes. Primes and the sieve of Eratosthenes. The Fundamental Thm of Arithmetic.	Assignment 2 Solutions
September 22 - 26	The Fundamental Thm of Arithmetic, cont'd. (Quiz 1) Equivalence relations. Congruences.	Assignment 3 Solutions
September 29- October 3	Congruences - cont'd. Fermat's little theorem, computing and solving equations in Z_n. Public Key crypto and RSA.	Assignment 4 Solutions	Quiz 1, October 1. 19:00 -- 20:30 in STBIO S1/4. Please be sitted at 18:50.
October 6-10	The ring of polynomials over a field F. Degree. Division with residue. GCD's. The Euclidean algorithm for polynomials. Irreducible polynomials.	Assignment 5 Solutions
October 15-17	Unique factorization. Roots of polynomials. Roots of rational and real polynomials.		No class October 13 (Thanksgiving)
October 20-24	Roots of polynomials over Z_p. (Quiz 2) Rings (recall). Ideals. Z and F[x] are principal ideal rings.	Assignment 6 Solutions
October 27-31	Homomorphisms and kernels. Quotient rings. First isomorphism theorem. F[x]/(f(x)).	Assignment 7 Solutions In question 11, it should say "Prove that d is not a square of a rational number" and "Prove that Q[\sqrt{d}] = .... is a subring of C and is in fact a field"	Quiz 2, October 29. 16:35-17:55 STBIO S1/4.
November 3-7	Constructing finite fields. Roots in extension fields. Chinese remainder theorem. Applications of CRT.	Assignment 8 Solutions	Deadline to submit assignment 7 is extended to Wednesday November 5.
November 10-14	Applications of CRT. (Quiz 3). Groups: the basic definition and examples. The symmetric group.	Assignment 9 Solutions
November 17-21	The dihedral group. Cosets and Lagrange's theorem. Homomorphisms and isomorphisms. Group actions on sets: first definitions and properties.	Assignment 10 Solutions	Quiz 3, November 19. 16:35-17:55 STBIO S1/4
November 24-28	Group actions on sets: Examples. Cauchy-Frobenius formula. Applications to Combinatorics.
December 1-2	Homomorphisms, normal subgroups, quotient groups and the first isomorphism theorem. Examples.

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, I think.
Specifically:

Part I consists of 20 multiple choice questions. Each question has a unique correct answer. There is no penalty for wrong answers.

Part II consists of 4 proof questions; you have to answer 3 out of 4.
Part III is a mandatory question.

The final covers all the material covered in class and the course notes, excluding the following sections: 17.6 (Eisenstein's Criterion), 24 (prime and maximal ideals), 32 (Cauchy's theorem). In sections 33.4 - 33.6, I only require you to know what we've covered 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 covered in the later parts of the course. Very heavy emphasis on groups, constructing fields and working in them, and congruences, group actions on sets. Though, again, this does not mean that other topics are not covered in the final!

Special Office hours prior to the exam: (on the week of December 1 - 5)

Day	Time
Monday (Eyal)	14:00 - 15:00
Tuesday (Mike)	09:00 - 10:30 Cancelled due to last minute emergency
Thursday	Victoria 09:00 - 11:00 Vicotria 16:00 - 18:00
Friday	Eyal 10:00 - 12:00 Mike13:00 - 15:00