Courses 2002-03  Algebra II

Algebraic Number Theory
(Topics in Number Theory I, MATH-726)
This course is open to advanced undergraduates, who have successfully completed the courses MATH 370/371 and with my approval.
Time: Thursday 08:30-10:30 (BURN 920), Friday 1:30-3:30 (BURN 1205)
Office hours: Monday 14:30-15:30, Wednesday 9:30-10:30, Friday 10:00-11:00.
Course syllabus: This is a first course in algebraic number theory. Algebraic number theory is very much the study of a special class of rings, called Dedekind rings. One can carry out arithmetic in such rings and use this knowledge to study classical Diophantine questions such as Pell's equation x2 + Dy2 = 1, Fermat's equation xn + yn = zn, which numbers are sum of squares etc. For this purpose we discuss the following notions:
- Integral elements and integral closure.
 - Localization and local properties.
 - Dedekind domains.
 - Valuations and completions.
 - Decomposition and ramification.
 - Trace and norm. 
- Discriminant and different.
 - Inertia and decomposition groups.
 - Class groups and Minkowski's bound.
 - Units and Dirichlet's theorem.
 - Fields of low degree.
 - Cyclotomic fields.
Text Books:
- Janusz,  Algebraic number fields. Second edition. Graduate Studies in Mathematics, 7. AMS. (Recommended as main text book).
- Frohlich and Taylor, Algebraic Number Theory, Cambridge studies in advanced mathematics, 27.
- Lang, Algebraic Number Theory, Springer LNM 110.
Note: It is not absolutely necessary to have the text books. Your class notes should suffice. The books are excellent, though.
Course pre-requisites: The course presupposes knowledge of rings, fields and tensor products of modules over a commutative ring and basic Galois theory at the level of  MATH 370/371. This background can be obtained from textbooks by an advanced undergraduate student wishing to take the course. The course is open to undergraduates with my approval. Apart from the prerequisites, the course will be self contained; we shall develop all the machinery that we need.
Course requirements: The grade will be based on a few assignments (15%) and a take-home exam (85%).
Take Home Exam

Algebra II   (Math 251)

Lecturer: Eyal Z. Goren  Office: BURN1108    Tel: 398-3815
Office Hours:  Mon 10:30-11:30,     Wed 10:30-11:30 and 16-17.

Special Office Hours for April 11-30:

(Usual office hours are cancelled)
Goren: April 24  8:30-10:30, April 25  8:30-10:30.
D'amours: April 28  14:00 - 16:00,  April 29  14:00 - 16:00.
(plus you can try and catch D'amours any other time for questions).
You can pick up assignments 10, 11 and old assignments too from D'amours after April 15.

TA: Martin D'amours.  Office: BURN 1036    Office hours: Mon 15:30-16:30, Fri 1:30-2:30.
Tutorial hours: Mon 14:35 -15:25  ARTS W20, Wed 17:00-18:00, BURN 920. (Note: those are two sessions, covering the same material, that are optional though highly recommended.)
Marker: Thomas Cocolios.

Time of Course: MWF 8:30-9:30 Lecture Hall: Burnside Hall  1B24
Dates of exams:
Midterm: Wednesday, March 5, 8:35-9:25.
Final: Wednesday  April 30 - 14:00 GYM.
  • Part I: 7 multiple choice questions, each worth 4 points. I hope you can do each in 5 minutes or less. Often you would have to choose the most correct answer. For example:    Let x, y be real numbers.   When is xy>0?    a) x, y are positive; b) x, y are non-zero and have the same sign; c) x=y = 1. d) x = y = -1.     I expect you to choose b) in this case.
  • Part II: Choice of 3 out of 4 questions. Each question is worth 24 points. Each question has 2-3 subquestions on usually different topics. The distribution of points over subquestions is equal. I hope you can do each question in 45 minutes.
  • Everything we did in class and in the assignments. 
  • You have to know all the proofs. Expecting to come up with your own proofs is naive. You have to memorize the proofs. 
  • There will be propositions to prove that we didn't do in class or the assignments. 
  • There will be numerical calculations to do, similar to those we did in assignments. Knowing how to find linear transformations with certain properties, to find a special form of operators (Primary Decomposition, Jordan, diagonal, diagonal in an orthonormal basis) and how to apply these special forms is a must. Practice that a lot. 
  • I expect you to know the material and how to use it intelligently. That requires doing many exercises, attemting past exams, reading the solutions of the assignments. A very good practice is to study together and try and invent questions and solve them.

Method of Evaluation:  *Weekly assignments (15%): Best 8 out of 11. Handed out and collected on Fridays. 
*Midterm exam (20%, unless Final exam is better!).
*Final exam (65%, unless better than midterm and in that case 85%). 
*Note carefully: The grade on the assignments is always 15% of the final grade, even if midterm or final exams are better. In case the right to write a supplemental exam is granted to a student, it would carry the same weight as the original final exam (i.e., 65%, unless better than midterm and in that case 85%), unless special circumstances arise. In that case, the student should approach me. 
Submitting and getting back your assignments:When handing out your assignments, put only your student number on it (and not your name) and, of course, the course name. I will hand out assignments in class on Friday, but you can also download them from the web (usually before). You can submit your assignments until Friday 12:00 at the main office on the 10th floor. I will handout the marked assignments on Fridays after class and give the TA those not claimed.
About the marking of assignments: The marker will check carefully only selected exercises (you will not know which before hand) and your grade will be based on the total number of questions you solved correctly and a more detailed evaluation of your performance in the selected questions.
    In case you believe that you should get a higher mark on your assignment, for whatever reason, first  resubmit your assignment with an explanatory note. If you are not statisfied with the answer, then come to see me. Only in very rare cases I will overrule the marker's decision.

Syllabus: Vector spaces, spanning sets, basis, dimension. Linear maps and their matrix representations. Applications to systems of linear equations. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Inner product spaces. Diagonalization of self-adjoint operators.
Text book: Lipschutz, Seymour/Schaum's outline of theory and problems of linear algebra, 3rd edition.
Additional references: Hoffman & Kunze/Linear Algebra, (Prentice Hall) ;   Lawson/Linear Algebra, (Wiley).

The main challenges of the course. (1) Writing rigorous proofs. There is a strong emphasis in this course on complete proofs of almost all result. The language is very formal and precise. Students are required to be able to right rigorous proofs. Recommendation: mimic the style of the lecturer (i.e. me), at least initially. A proof should state clearly the given facts and the theorems assumed known and proceed from there by logical deductions, that are  clearly stated, to the conclusion one seeks.  (2) The concepts of a vector space, basis, dimension and of a linear transformation. The challenge is in digesting the abstract concepts of vector spaces and linear transformations in a "coordinate free" approach.  Recommendation: this requires deep thinking and having many examples at your fingertips. One should definitely allow time just to digest the concepts and intuition will be created on the basis of many examples. (3) Applying the abstract theory in concrete problems. The syllabus compels us to spend most of the lectures on definitions, theorems and proofs. We shall try, of course, and supply examples and applications in class. However, most of the applications and problems requiring calculations will be given in the assignments. The challenge would be to see the applicability of the theorems we prove in class to concrete situations. Recommendation: come to tutorials and office hours. Attempt additional questions in the text book (which contains solutions).
A general recommendation. Study with other people, preferably neither stronger nor weaker than you. However, in the end of the day, each person has to right his or her own solutions to the assignments.

Detailed Syllabus
Week  Material  Assignment  Comments and Corrections Learn More! (No... it's not on the final!) 
January 6-10 Motivation for studying vector spaces, linear map and matrices. Defn of a v.sp., subspace and examples. Sum and direct sum of subspaces. Linear dependence, span and spanning set. Assign. 1  pdf
Assign. 1  dvi

Sol'ns 1  pdf
Sol'ns 1  dvi

Use textbook for more examples and careful verification of the axioms for them. 
Minor Corrections: 
min{d(x; 0) : x C; x ¹ 0}
A linear code C corrects t errors if and only if the Hamming distance of every two distinct elements of C is at least 2t + 1.
History of abstract linear spaces.
(linear spaces = vector spaces).
January 13-17 Steinitz's substitution lemma, basis, dimension, dim (W+U). Basis, coordinates, change of basis. Assign. 2  pdf
Assign. 2  dvi

Sol'ns 2  pdf
Sol'ns 2  dvi

January 20-24 Linear maps: def., Ker and Im, examples. Non-singular maps, isomorphism, composition. dim(Ker)+dim(Im) = dim(V). dim(U+W)  - another proof. Nilpotent operators.Fitting's lemma. Assign. 3  pdf
Assign. 3  dvi

Sol'ns 3  pdf
Sol'ns 3  dvi

January 27-31 Projections. Linear transformations and matrices. Examples. Determinants. Assign. 4  pdf
Assign. 4  dvi

Sol'ns 4  pdf
Sol'ns 4  dvi

In question 7 read 'permutation' instead of 'transformation' hand out on permuations
February 3-7 Determinants. Existence and Uniqueness, multiplicativity, Laplace's Theorem, the adjoint matrix. Linear equations. Assign. 5  pdf
Assign. 5  dvi

Sol'ns 5  pdf
Sol'ns 5  dvi

February 10-14 Linear equations: row space and column space; reduced echelon forms; row-rank = column rank; Cramer's rule. Assign. 6  pdf
Assign. 6  dvi

Sol'ns 6  pdf
Sol'ns 6  dvi

February 17-21 Inner product spaces; metric and norm; Cauchy-Schwartz inequality; the triangle inequality; positive definite Hermitian matrices; orthonormal bases and Gram-Schmidt process; orthogonal projection; the method of least squares. Assign. 7  pdf
Assign. 7  dvi

Sol'ns 7  pdf
Sol'ns 7  dvi

There is a typo in the deadline for submitting. You can submit ass. 7 until March 7, 12:00.
March 3-7 Eigenvalues and eigenspaces. The characteristic polynomial. Algebraic and geometric multiplicity. MIDTERM ON MARCH 5
Everyone should be sitted by 8:30.
Solutions to Midterm: dvi or pdf
The midterm has 3 questions. One is to prove a theorem proved in class; the other is to prove a couple of easy propositions you didn't see before; the last is computational question of the kind appearing in the assignments.
March 10-14 Diagonalization. Application to recursion sequences. Criterion: m_g = m_a for all e.values; Cayley-Hamilton; the minimal poly. and the char. polyl. Diagonalization and the min. poly.  m_A|D_A|m_A^n Assign. 8  pdf
Assign. 8  dvi

Sol'ns 8  pdf
Sol'ns 8  dvi

March 17-21 The Primary Decomposition Theorem. Diag'able iff m_A is a product of distinct lin. factors. Jordan canonical form. Assign. 9  pdf
Assign. 9  dvi

Sol'ns 9  pdf
Sol'ns 9  dvi

Example given in class
In Q. 3, Assign. 9, assume that the char. of the field is not 2 (so every non-zero scalar is has two distinct roots).
March 24-28 Jordan canonical form (cont'd). Self-adjoint operators.  Assign. 10  pdf
Assign. 10  dvi

Sol'ns 10  pdf
Sol'ns 10  dvi

Example given in class

Another Example given in class

March 31- April 4 Self-adjoint operators (cont'd). Application to inner products. Normal operators and the Spectral Theorem.  Assign. 11  pdf
Assign. 11  dvi

Sol'ns 11  pdf
Sol'ns 11  dvi

Another example of JCF
April 7 - 11 The dual space. Bilinear forms.
Final Exam     Solutions (Sketch)