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 8:359:25 in the basement of
TA's: Dylan AttwellDuval, Luiz Takei.
Tutorial Hours:
Mon 16:35  17:55 (AttwellDuval) in BURN1214
Tue 16:35  17:55 (Takei) in BURN1214
TA office hours:
AttwellDuval: Office 1018, Mon 15:0016:00 and Fri
15:0016:00
Takei: Office 1033, Mon 11:0012:00 and Tue
15:0016:00
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):
Evaluation method:
Academic integrity:
Submitting work. In accord with
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 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. 

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 24 
Sets, Methods of Proof. 


September 711 
Functions. On the notion of cardinality. 
Monday is Labour day
 no classes 

September 1418 
Complex numbers.
Polynomials and the fundamental thm of Algebra. 


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. 


September 28 October
2 
Applications of the fund’l thm. Equivalence relations. Congruences. (Quiz 1) 


October 59 
Fermat's little theorem,
computing and solving equations in Z_{n}. Public Key
crypto and RSA. 
Quiz 1,
Tuesday October 6, 17:35 

October 1216 
The ring of polynomials
over a field F. Degree.
Division with residue. GCD's. The Euclidean
algorithm for polynomials. 
Monday is Thanksgiving
day (no class) 

October 1923 
Irreducible polynomials. Unique factorization. Roots of polynomials.
Roots of rational and real polynomials. Roots of polynomials over
Z/pZ. 
No
assignment this week 

October 2630 
Ideals. Z
and F[x] are
principal ideal rings. (Quiz
2)
Quotient
rings. First isomorphism theorem. F[x]/(f(x)). 


November 26 
F[x]/(f(x)), continued. Constructing finite
fields. Roots in extension fields. (Quiz 3). Homomorphisms and kernels. Chinese remainder
theorem. 
Quiz 2, Tuesday November
3, 17:35 Maass 112 and Maass 217 

November 913 
Applications of CRT.
Groups: the basic definition and examples. 


November 1620 
The symmetric group. The dihedral group. Cosets
and Lagrange's theorem. Group actions on sets: first
definitions and properties. 
Quiz 3, Tuesday November
17, 17:35 Adams Aud 

November 2327 
Group actions on sets:
Examples. CauchyFrobenius
formula. Applications
to Combinatorics. 


November 30 
December 2 
Homomorphisms and isomorphisms, Cayley's theorem,
normal subgroups,
quotient groups and the first isomorphism theorem. Examples. 

