Courses 2004-05                            last updated: December 31, 2004.

Algebra 3, MATH 370. Algebra 4, MATH 371
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Algebra 4, MATH 371
Instructor: Dr. Eyal Z. Goren.
Time: MWF 8:35-9:25    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 . My office is Burnside Hall, 1108; Tel: 514-398-3815.
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.
February-April: 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.
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.        (Handed-out 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.

Issues of 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 for more information).

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. 356-358. 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

2/28 Cyclotomic fields (cont'd). 
Galois Theory.
The automorphism group and subfields. 
Assignment 6
3/7 Definition of a Galois extension. A splitting field is Galois. Examples. Linear independence of characters. K/KG 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 1-11 in Section 14.3
Exercises 1-6 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 S5 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.

Algebra 3, MATH 370

Time: MWF 8:35-9:25    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):
September-October: 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: semi-direct products, the simplicity of An, solvable groups.
November-December: 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.
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 text-book for MATH 371).
Other texts:
M. Artin/ Algebra.
J. Rotman/ Introduction to the theory of groups.
Evaluation Method:
* 10% Assignments (11 weekly, short assignments. Handed-out 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 9:30-11:00. (Burnside Hall, 1108; 514-398-3815).

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 D2n    
9/8  The Symmetric group Sn (cycles, sign, transpositions, generators). The group GLn(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
9/20  Homomorphism, kernels and normal subgroups. The homomorphism theorems.   Assignment 2
9/27 The lattice of subgroups. Group actions on sets: actions, stabilizers and orbits. Examples. Cayley's theorem. The Cauchy-Frobenius formula.    Assignment 3
Suplement to Solutions
10/4 Applications to combinatorics: necklaces designs, 14-15 square, Rubik's cube. Conjugacy classes in Sn. Conjugacy classes in An.The simplicity of An  In Q. 3 it should read (q^2 + 3q + 2f)/6 Assignment 4
10/13 The class equation. p-groups. Free groups and Burnside's problem.  10/11 no class - Thanksgiving.  Assignment 5
10/18 Syllow's Theorems and applications (e.g., groups of order pq and p2q). Direct and semi-direct products.
Assignment 6
10/25 Semi-direct products (continued). Finitely generated abelian groups (statement only). Groups of order less than 16.  These are proof-read solutions. Including a comment on the problematic question and correction of other typos. Assignment 7
11/1 Composition series. The Jordan Holder Theorem. Solvable groups. Rings - basics. 
Assignment 8
11/8 Ideals and quotient rings. Examples: Z, Z/nZ, R[x], R[[x]], R((x)), Mn(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
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
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.   

The following does not reflect in any way my opinion on your integrity. It must be included by University regulations:
Issues of 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 for more information).