Courses 2003-04
Vectors, Matrices and Geometry, MATH 133.                                        last updated: May 2, 2004.
Algebra 4, MATH 371.
Algebra 3, MATH 370.

Vectors, Matrices and Geometry, MATH 133.

Instructor: Dr. Eyal Z. Goren.
Time: MTWThF 9:05-10:55    Location: BURN 1B23
Syllabus (section numbers refer to Poole's book): Geometric vectors, dot product, lines and planes (Sections 1.1, 1.2, 1.3, Cross Products); Systems of linear equations, spanning sets and linear independence (Sections 2.1, 2.2, 2.3); Matrices, matrix algebra, subspaces, basis, dimension, rank, linear transformations (Sections 3.1, 3.2, 3.3, 3.4, 3.5); Eigenvalues and eigenvectors, determinants, similarity and diagonalization (Sections 4.1, 4.2, 4.3, 4.4); Orthogonality, orthogonal complements and orthogonal projections, the Gram-Schmidt process. (Sections 5.1, 5.2, 5.3,5.4, 5.5(Quadratic Forms)).
Prerequisite: A course in functions. See Calendar for additional information.
Text Book:  Linear Algebra, by D. Poole.
Additional Texts: The text book will suffice for everything in this course. You may however consult also the following texts if you need.
1) Anton, H. "Elementary linear algebra", 7th ed., or Anton & Rorres "Elementary linear algebra, applications version", 8th ed., Chapters 1-7.
2) Nicholson, W. K., "Linear algebra with applications", 2nd ed., Chapters 1-7.
Evaluation Method:
* WeBWorK assignments  8%. There will be 4 Assignments posted every Thursday with a deadline of Tuesday. Your WeBWorK assignments are on  http://msr01.math.mcgill.ca/webwork/m133504/
* Written assginments  7%. There will be 2 such Assignments to be given on May 17 and May 25 and handed in by Friday May 22 and May 28, respectively. Solutions to Midterm.    Midterm Grades (out of 100)
* Midterm exam (compulsory)  15%. To be given on Monday, May 17,  14:00-16:00. Room  ARTS W-120. Please verify this information and bring your McGill ID to the exam. The midterm will consist of one proof we did in class, one easy proof not done in class and exercises as in the assignments.
* Final Exam  70%. To be given on Friday, May 28, 9:00-12:00, Room: LEA 14 & 15. Please verify this information and bring your McGill ID to the exam. The structure of the final will be similar to the midterm's; there will be some easy proofs, but most of the exam will consist of exercises very similar to those in the assignments and examples done in class. Final Exam Copy
Office Hours: Tue, Th  11:00-12:00. (Burnside Hall, 1108; 514-398-3815).
Help Desk: There is a "help desk" which is a study center where advanced undergrads and graduate students are at hand to assist you. This is highly recommended.

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 www.mcgill.ca/integrity for more information).

Detailed Syllabus
(§ numbers refer to Poole's book)
 Date Material Comments Written Assignments Handouts May 3-7 Vectors: geometric and algebraic; adding and substracting vectors.  Dot product; lengths and angles.  Lines and planes. Cross product. Linear equations. Methods for solving linear equations.  Spanning sets and linear dependence. §§ 1.1-1.3, 2.1-2.3 Students not familiar with complex numbers should read Appendix C in Poole. Knowledge of complex numbers is assumed from the fourth lecture. Solving linear equations New Version May 12 May 10-14 Algebra with matrices: addition, multiplication by scalar, multiplication, transpose. Elementary matrics. The inverse matrix and its calculation by row-reduction; application to linear equations. Subspace; row, column and null space of a matrix; basis and dimension. Linear transformations. Linear transformation and matrices. Rotations. Reflections. Composition. §§ 3.1-3.5 Algebra with matrices May 17-21 Projections. Inverse linear transformation. Eigenvalues and eigenvectors. Determinants. Laplace's expansions. Determinants of elementary matrices. The product (and other) formula for determinants. Calculation of determinants by row reduction. Determinant as a volume function. Cramer's rule and the adjoint matrix. The characteristic polynomial. Eigenvalues and eigenvectors revisited. Linear independence of eigenspaces. Similarity and diagonalization. Representing linear transformations in different bases and diagonalization. Applications. §§ 4.1-4.4 Written Ass 1 May 24-28 Orthogonality in Rn. Orthonormal bases. Orthonormal matrices and distance preserving transformations. Orthogonal complements and orthogonal projections. The Gram-Schmidt process. A Symmetric matrice has real e.values and its e.spaces are orthogonal to each other. Orthogonal diagonalization. Applications: Quadratic forms and extrema of functions of 2 variables. §§ 5.1-5.5 May 24 is Victoria Day; make-up class is given ub 1B23 on May 25 and 26, 11:30 - 12:25. Written Ass 2 Diagonalization algoirthms Notes for Wednesday Lectures

Algebra 4, MATH 371

Instructor: Dr. Eyal Z. Goren.
Time: MWF 8:35-9:25    Location: BURN 920.
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. Time permitting: Smith's normal form.
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: Infinite Galois groups. 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).
Note: This course is normally taken by honours students, though I do not consider that a requisite. One can therefore expect it to be exciting, inspiring but also challenging.
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:
* 20% Assignments    (12 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.)
* 25% Midterm         (Tuesday, March 9, 17:30 - 19:00  BURN 1120). 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.
* 55% Final Exam     (Tuesday, April 20, 14:00, MAASS 328)
-- If final exam grade is better than midterm then midterm doesn't count. Assignment grades always count (even in deferred/supplamental).
Office Hours: Wednesday, Friday  9:30-11:00. (Burnside Hall, 1108; 514-398-3815).

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 www.mcgill.ca/integrity for more information).

Detailed Syllabus

(§ numbers refer to Dummit and Foote)
 Date Material Comments Assignments and Solutions 1/5 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. §§ 10.1-10.2 1/12 Chinese Remainder Theorem. Exact sequences. 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. §§ 10.2-10.3, § 12.1 Assignment 1 1/19 The elementary divisors theorem (cont'd). The structure theorem for  f.g. modules over a PID. §§ 12.1-12.2 Assignment 2 1/26 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, F[x]/f(x), F(a); §§ 12.3, 13.1 Assignment 3 2/2 Degree, algebraic elements, algebraic extensions. §§ 13.1 - 13.2 Assignment 4 2/9 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. § 13.3 Assignment 5 2/16 Splitting fields and algebraic closure (cont'd). Finite fields. Cyclotomic fields. §§ 13.4 - 13.6 Assignment 6 2/23 STUDY BREAK 3/1 Cyclotomic fields (cont'd).  Galois Theory. The automorphism group and subfields. Assignment 7 3/8 Mideterm on March 9, 17:30-19:00, BURN 1120. Definition of a Galois extension. A splitting field is Galois. Examples. Linear independence of characters. K/KG is Galois with Galois group G. Assignment 8 3/15 A Galois extension is a splitting field of a separable polynomial. The fundamental theorem of Galois theory. Assignment 9 3/22 Examples (finite fields and cyclotomic fields revisited). Construction of regular polygons. An S5 extension. Composite extensions. Assignment 10 3/29 The primitive element theorem. C is algebraically closed. Solvable and radical extensions. The insolvability of the quintic. Assignment 11 4/5 Galois groups of polynomials and the calculation of Galois groups over Q. 4/13 Galois groups of polynomials and the calculation of Galois groups over Q (cont'd).

Bonus problems.
You may solve and submit these problems at any time before the last lecture. This will improve your assignment grades. Partial solutions are allowed. Re-submittion is allowed.

1. Give an explicit example of a real positive number  r  such that [Q(r):Q] is a power of two but  is not constructible.
2. Let F = Z/pZ a field with p elements and K an algebraic closure. Find all subfields of K.
3. Give an example of a polynomial f(x) in Z[x] such that f(x) is irreducible but f(x) is reducible modulo p for every prime p.
4. Let p>2 be a prime. Prove that the unique quadratic field K (i.e. [K:Q] = 2) contained in Q(exp(2pi/p)) is Q(t1/2) where t = (-1)(p-1)/2p.

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 we'll 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.
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 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 Monday, November 3. Time and Place: 17:05-18:25, ARTS 210 )
* 70% Final Exam.    (to be conducted on Thursday, December 18, 9:00 - 12:00, in ARTS W-20)
-- If final exam grade is better than midterm then midterm doesn't count. Assignment grades always count (even in deferred/supplamental).
Office Hours: Monday, Wednesday, Friday 10:00-11:00. (Burnside Hall, 1108; 514-398-3815).
Special Office hours prior to the exam: I will be away on lectures and conferences during December 4 - 8 (until noon) and 13-19. However, during December 9 - 12, I will see students each day from 10:00 - 11:00.

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 www.mcgill.ca/integrity for more information).

Detailed Syllabus

 Date Material Comments Assignments and Solutions 9/3 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. Cyclic groups and the structure of their subgroups. The group F* is cyclic. Commutator, centralizer and normalizer subgroups. Cosets. Refresh your memory of the symmetric group. Assignment 1 Solutions 9/15 Cosets. Lagrange's Theorem. Normal subgroups and Quotient groups. Abelianization. Homomorphism, kernels and normal subgroups. The first homomorphism theorem. In question 3) (2), p is a prime. Assignment 2 Solutions 9/22 The homomorphism theorems (cont'd). The lattice of subgroups. Group actions on sets: actions, stabilizers and orbits. Examples. Assignment 3 Solutions 9/29 Group actions on sets (cont'd): Cayley's theorem. The Cauchy-Frobenius formula. Applications to combinatorics: necklaces designs, 14-15 square, Rubik's cube. Conjugacy classes in Sn. Assignment 4 Solutions 10/6 Conjugacy classes in An.The simplicity of An. The class equation. p-groups. In question 1, the group G acts linearly on the vector space V. Assignment 5 Solutions You can hand in your assignment 5 on Wednesday October 15. 10/13 Free groups and Burnside's problem. Cauchy's Theorem. Syllow's Theorems -- statement and examples. Assignment 6 Solutions 10/20 Syllow's Theorems -- proof and applications (e.g., groups of order pq and p2q). Finitely generated abelian groups. This version --> of the assignment correct typos of the one given in class. Assignment 7 Solutions Number of Groups of  order N 10/27 Semi-direct products and groups of order pq. Groups of order less than 16. Composition series. The Jordan Holder Theorem. Assignment 8 Solutions 11/3 Solvable groups. Rings - basics. Ideals and quotient rings. Examples: Z, Z/nZ, R[x], R[[x]], R((x)). Midterm on Monday, November 3 17:05-18:25, ARTS 210 Midterm Solutions Midterm Grades 11/10 Examples: Mn(R), Quaternions. Creating new rings: quotient, adding a free variable, field of fractions. Ring homomorphisms. First isomorphism theorem. Behavior of ideals under homomorphisms. In question 2, part (1), assume R is an integral domain! Assignment 9 Solutions 11/17 More on ideals: intersection, sum, product, generation, prime and maximal. The Chinese Remainder Theorem. Euclidean rings. Examples: Z, F[x], Z[i]. PID's. Euclidean implies PID. Greatest common divisor and the Euclidean algorithm. Assignment 10 Solutions 11/24 The Euclidean algorithm. Prime and irreducible elements + agree in PID. UFD's. Prime and irreducible agree in UFD. PID implies UFD. g.c.d. in a UFD. Gauss's Lemma. Assignment 11 Solutions 12/1 R UFD implies R[x] UFD. Existence of splitting fields. Construction of finite fields.