Algebra
3, MATH 370. Algebra
4, MATH 371
The URL of this page is
http://www.math.mcgill.ca/goren/courses0405.html
Algebra
4, MATH 371
Instructor: Dr. Eyal Z. Goren.
Time: MWF 8:359:25
Location: BURN 920.
Office Hours: Friday 9:30  10:00
and by appointment (I will be available for meeting on every day of the
week; you can also set an appointment by email to goren@math.mcgill.ca)
. My office is Burnside Hall, 1108; Tel: 5143983815.
Syllabus (in the large):
We shall cover part of Chapter 10 and
most of chapters 12, 13, 14 of Dummit and Foote.
January: Introduction to modules.
Modules over PID. Applications to linear transformations and finitely
generated
abelian groups.
FebruaryApril: Introduction to
filed theory. Algebraic and transcendental extensions; separable and
inseparable
extensions. Splitting fields and algebraic closure. Galois groups. The
fundamental theorem of Galois theory. Applications to solving equations
by radicals. Finite and cyclotomic fields. Time permitting: The
inverse Galois problem.
Prerequisite:
MATH 251, MATH 370 (or equivalent courses
with my permission. Students that haven't taken a course on vector
spaces
can still enrol, given my permission, but are advised to catch up on
this
material by reading Dummit and Foote Sections 11.1  11.4).
Text Book: *
Dummit and Foote/ Abstract Algebra (Third Edition), Wiley.
Other texts (on reserve at Schulich):*
M. Artin / Algebra. *
S. Lang / Algebra. *
N. Jacobson / Abstract algebra *
I. Stewart / Galois Theory
Evaluation Method:
*15%
Assignments. (Handedout
and submitted on Mondays. Submit all. You may work together on your
assignment,
but in the end each has to write his or her own solutions; identical
assignments
will be marked as zero.)
* 10%
quiz number 1. The topics are module
theory
including modules over PID. To be conducted in class January 31,
9:00  9:30 (in class). You will be asked to prove a statment we proved
in class (or provide part of a long proof given some intermediate
lemmata)
and to solve an exercise from Dummit & Foote. Results.
The
Quiz.
*20%
Midterm.
The topics are General Theory of Modules, Modules over a PID and
General
Theory of Fields. The material includes those parts of Chapters 10 and
12 covered in class, and sections 13.1  13.4, 13.6 in Chapter 13 of
Dummit
and Foote. See detailed syllabus below.To
be conducted on Thursday, March 3, 16:00  18:00 in Burnside Hall Room
708. Results.
The
Midterm.
* 5%
quiz number 2. On the material
from
the start of the topic of Galois theory up to and including the proof
of
the Main theorm of Galois theory.
To be conducted on Wednesday, March 23, 9:00  9:25.
Results. The
Quiz.
*50%
Final Exam.
* The
calculation of the final grade: it is the better from the grade
composed as above and the grade of the final exam alone.
Detailed Syllabus
(§ numbers refer
to
Dummit and Foote. Dates are approximate and will be updated during the
semester)
Date  Material  Comments/suggestions  Assignments and Solutions 
1/4  General
Theory of Modules: Modules, submodules, morphisms of modules. Examples (in particular with base ring Z or F[x]). Kernels and quotient modules. The isomorphism theorems. 
We are doing now D&F 10.1  10.2. Besides finishing all details I leave to you during class (many of which appear as exe. in D&F), you can try and do Exe. 4, 7, 8, 10, 11, 13, 19 in section 10.1 and Exe. 3, 4, 5, 6, 9, 10, 12, 13 in section 10.2.  . 
1/10  Chinese
Remainder Theorem.
More examples. Free modules. Torsion and rank. Sum and direct sum of
modules. Modules over a PID: The elementary divisors theorem for f.g. modules over a PID. 
We are doing now D&F 10.3, 12.1. Recommended exercises are Ex. 5, 7, 9, 10, 11, 12, 18, 19, 22, 27 on pp. 356358.  Assignment 1 
1/17  The elementary divisors theorem (cont'd). The structure theorem for f.g. modules over a PID.  We are doing now D&F 10.3, 12.1  12.3. Recommended exercises are in Section 12.1, Exe. 1, 2, 3, 4, 5, 6, 9, 17, 18, 19. Section 12.2 Exe. 3, 4, 10, 11, 12, 15, 22, 23, 24, 25  Assignment 2 
1/24  The structure
theorem for
f.g. modules over a PID (cont'd). The rational canonical form.
Application
to Jordan's canonical form. General Theory of Fields: Fields  main examples, characteristic. 
I suggest
in section
12.3 Exe. 1, 17, 19 (most of the rest are very specifically about the
Jordan
form and its applications which I assume you've seen in MATH251).
We started section 13.1 
Assignment 3 
1/31  F[x]/f(x), F(a), Degree, algebraic elements, algebraic extensions.  We
continue section
13.1 and start with 13.2 I recommed trying 13.1 Exe. 1, 2, 3, 4, 5, 6, 7, 8. 
No Assignment! 
2/7  Algebraic extension (cont'd). Straighthedge and compass constructions. The negative solution to doubling the cube, trisecting an angle, quadrature of the circle. Splitting fields and algebraic closure.  We do sections 13.2, 13.3. I recommend 13.2 Exe. 4, 5, 6, 8, 13, 14, 17, 18, 20.  Assignment 4 
2/14  Splitting fields and algebraic closure (cont'd). Finite fields. Cyclotomic fields.  Section 13.4, 13.6 + material on finite field which is scattered in the text book. I recommend 13.4, Exe. 1, 2, 3, 4;  Assignment 5 
2/21  STUDY BREAK  
2/28  Cyclotomic
fields (cont'd). Galois Theory. The automorphism group and subfields. 
SOLUTIONS
TO ASSIGNMENTS 
Assignment 6 
3/7  Definition of a Galois extension. A splitting field is Galois. Examples. Linear independence of characters. K/K^{G} is Galois with Galois group G.  The material in Galois theory is selected material from 14.1
 14.8 Here are exercises you may wish to do for practice: All Exercises in sections 14.1, 14.2 are suitable. Exercises 111 in Section 14.3 Exercises 16 in Section 14.4 All exercises in Section 14.5 Be careful with the exercises in Section 14.6. Some assume material we didn't discuss. 
Assignment 7 
3/14  A Galois extension is a splitting field of a separable polynomial. The fundamental theorem of Galois theory.  Assignment 8 

3/21  Examples (finite fields and cyclotomic fields revisited). Construction of regular polygons. An S_{5} extension. Composite extensions.  Assignment 9  
3/28  Composite
extensions. The
primitive
element theorem. 

4/4  C is algebraically closed.Galois groups of polynomials and the calculation of Galois groups over Q.  Assignment 10  
4/11  Solvable and radical extensions. The insolvability of the quintic. 
Time: MWF 8:359:25
Location: BURN 920.
Course Notes:
Here
are notes in pdf. The notes will be updated during the semester and
expanded to include all the material in group theory and ring theory we
shalll cover in the course. Be aware that notes may still contain
typos;
please let me know if you find any!
Syllabus (in the large):
SeptemberOctober: Introduction
to groups, permutation groups; the isomorphism theorems for groups; the
theorems of Cayley, Lagrange and Sylow; structure of groups of low
order.
Time permitting: semidirect
products,
the simplicity of A_{n}, solvable groups.
NovemberDecember: Introduction
to ring theory; integral domains, fields, quotient field of an integral
domain; poynomial rings; unique factorization domains.
Time permitting: principal ideal
domains, Euclidean domains.
Prerequisite:
MATH 251 (or an equivalent course).
Note: This course is normally taken
by honours students, though I do not consider that a requisite
(approval
required). One can therefore expect it to be exciting, inspiring but
also
challenging.
Text Book:
Dummit and Foote/ Abstract Algebra
(Third
Edition), Wiley.
(Available at the bookstore; This is also
the textbook for MATH 371).
Other texts:
M. Artin/ Algebra.
J. Rotman/ Introduction to the theory
of groups.
Evaluation Method:
* 10%
Assignments
(11 weekly, short assignments. Handedout and submitted on Mondays.
Submit
all. You may work together on your assignment, but in the end each has
to write his or her own solutions; identical assignments will be marked
as zero.)
* 20%
Midterm
(to be conducted on Friday, November 5, 15:00  16:30 in Room BURN 1205)Midterm
Grades
Midterm Exam
* 70% Final
Exam. (to be conducted on December 21, 14:00 in
Burnside
Hall 1B39)
 If final exam grade is better
than midterm then midterm doesn't count. Assignment grades always count
(even in deferred/supplamental).
Office Hours: Monday 9:3011:00.
(Burnside Hall, 1108; 5143983815).
Detailed Syllabus (approximate)
Date  Material  Comments  Assignments and Solutions 
9/1  Introduction. Groups. Subgroups. Order of an element and the subgroup is generates. Subroup generated by a set. The groups Z, Z/nZ, Z/nZ*. The Dihedral group D_{2n}.  
9/8  The Symmetric group S_{n }(cycles, sign, transpositions, generators). The group GL_{n}(F). The quaternion group Q. Groups of small order. Direct products. The subgroups of (Z/2Z)^{2}.  9/6 no class  Labour day.  
9/13  Cosets. Lagrange's Theorem. Cyclic groups and the structure of their subgroups. The group F* is cyclic. Commutator, centralizer and normalizer subgroups. Normal subgroups and Quotient groups. Abelianization.  Assignment
1 Solutions 

9/20  Homomorphism, kernels and normal subgroups. The homomorphism theorems.  Assignment
2 Solutions 

9/27  The lattice of subgroups. Group actions on sets: actions, stabilizers and orbits. Examples. Cayley's theorem. The CauchyFrobenius formula.  Assignment
3 Solutions Suplement to Solutions 

10/4  Applications to combinatorics: necklaces designs, 1415 square, Rubik's cube. Conjugacy classes in S_{n}. Conjugacy classes in A_{n}.The simplicity of A_{n}.  In Q. 3 it should read (q^2 + 3q + 2f)/6  Assignment
4 Solutions 
10/13  The class equation. pgroups. Free groups and Burnside's problem.  10/11 no class  Thanksgiving.  Assignment
5 Solutions 
10/18  Syllow's Theorems and applications (e.g., groups of order pq and p^{2}q). Direct and semidirect products.  Assignment
6 Solutions 

10/25  Semidirect products (continued). Finitely generated abelian groups (statement only). Groups of order less than 16.  These are proofread solutions. Including a comment on the problematic question and correction of other typos.  Assignment
7 Solutions 
11/1  Composition series. The Jordan Holder Theorem. Solvable groups. Rings  basics.  Assignment
8 Solutions 

11/8  Ideals and quotient rings. Examples: Z, Z/nZ, R[x], R[[x]], R((x)), M_{n}(R), Quaternions. Creating new rings: quotient, adding a free variable, field of fractions. Ring homomorphisms.  
11/15  First isomorphism theorem. Behavior of ideals under homomorphisms. More on ideals: intersection, sum, product, generation, prime and maximal.  Question 4, (2) is bonus. Should have been Z[i]/(3) is a field.  Assignment
9 Solutions 
11/22  The Chinese Remainder Theorem. Euclidean rings. Examples: Z, F[x], Z[i]. PID's. Euclidean implies PID. Greatest common divisor and the Euclidean algorithm. Prime and irreducible elements + agree in PID. UFD's.  Question 1 should be: Prove that every prime ideal of Z[x] is of the form (n), (f(x)) or (n, f(x)). Provide an example in each case.  Assignment
10 Solutions 
11/29  Prime and irreducible agree in UFD. PID implies UFD. g.c.d. in a UFD. Gauss's Lemma. R UFD implies R[x] UFD. 