AlgebraI

Algebra 1, MATH235                              page last update: November 18, 2009.

Instructor: Eyal Goren
Office: BURN 1108.
Office hours: Mon 11:30 - 12:30, Fri 11:30 - 12:20.
Time and location of course: MWF 8:35-9:25 in the basement of Burnside Building 1B45

TA's: Dylan Attwell-Duval, Luiz Takei.
Tutorial Hours:
    Mon 16:35 - 17:55 (Attwell-Duval) in BURN1214
    Tue 16:35 - 17:55 (Takei) in BURN1214
TA office hours:
    Attwell-Duval: Office 1018, Mon 15:00-16:00 and Fri 15:00-16:00
    Takei: Office 1033, Mon 11:00-12:00 and Tue 15:00-16:00
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:

5% 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. Submitting assignments is a great way to get input on how you are doing in the course. The weight of the assignments in the final grade is such that it makes almost not worth the trouble of plagiarism. I hope that you submit only your own work, otherwise you would be missing the only chance to test your understanding of the course material in real time.
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.
---- Due to the H1N1 situation, I am going to calculate the Quiz component of the final grade as best 2 out of 3. Also, this should encourage you to study hard for Quiz 3. If you missed two Quizzes your final grade is determined by the final exam alone. ----
65% 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).
Submitting work. In accord with McGill University’s Charter of Students’ Rights, students in this course have the right to submit in English or in French any written work that is to be graded.
Syllabus and Grade Calculation. In the event of extraordinary circumstances beyond the University’s control, the content and/or evaluation scheme in this course is subject to change.

Grades of Quizzes		Remarks
Quiz 1	Results
Quiz 2	Results	* Quiz 2 was harder than Quiz 1, and the class performance was dismal. Quiz 2 was exactly at the level you should expect on the final. * Due to the H1N1 situation, I am going to calculate the Quiz component of the final grade as best 2 out of 3. Also, this should encourage you to study hard for Quiz 3. If you missed two Quizzes your final grade is determined by the final exam alone. * To see your Quiz 2, go to the TA's office hours. * There were some cases where we have recently decided to increase the grade. The new list reflects this.
Quiz 3	Results	To see your quiz see the TA's.

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 2-4	Sets, Methods of Proof.
September 7-11	Functions. On the notion of cardinality.	Assignment 1 Solutions	Monday is Labour day -- no classes
September 14-18	Complex numbers. Polynomials and the fundamental thm of Algebra.	Assignment 2 Solutions
September 21 - 25	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 3 Solutions
September 28- October 2	Applications of the fund’l thm. Equivalence relations. Congruences. (Quiz 1)	Assignment 4 Solutions
October 5-9	Fermat's little theorem, computing and solving equations in Z_n. Public Key crypto and RSA.	Assignment 5 Solutions	Quiz 1, Tuesday October 6, 17:35 A - L in LEA 26 M - Z in LEA 219
October 12-16	The ring of polynomials over a field F. Degree. Division with residue. GCD's. The Euclidean algorithm for polynomials.	Assignment 6 Solutions	Monday is Thanksgiving day (no class)
October 19-23	Irreducible polynomials. Unique factorization. Roots of polynomials. Roots of rational and real polynomials. Roots of polynomials over Z/pZ.	No assignment this week
October 26-30	Ideals. Z and F[x] are principal ideal rings. (Quiz 2) Quotient rings. First isomorphism theorem. F[x]/(f(x)).	Assignment 7 Solutions
November 2-6	F[x]/(f(x)), continued. Constructing finite fields. Roots in extension fields. (Quiz 3). Homomorphisms and kernels. Chinese remainder theorem.	Assignment 8 Solutions	Quiz 2, Tuesday November 3, 17:35 Maass 112 and Maass 217
November 9-13	Applications of CRT. Groups: the basic definition and examples.	Assignment 9 Solutions
November 16-20	The symmetric group. The dihedral group. Cosets and Lagrange's theorem. Group actions on sets: first definitions and properties.	Assignment 10 Solutions	Quiz 3, Tuesday November 17, 17:35 Adams Aud NOTE: no need to resubmit question 8. I forgot I've already given it.
November 23-27	Group actions on sets: Examples. Cauchy-Frobenius formula. Applications to Combinatorics.	Assignment 11 Solutions	DO NOT SUBMIT THIS ASSIGNMENT. IT IS FOR PRACTICE IN VIEW OF THE FINAL EXAM.
November 30 - December 2	Homomorphisms and isomorphisms, Cayley's theorem, normal subgroups, quotient groups and the first isomorphism theorem. Examples.