6 ECTS credits
150 h study time
Offer 1 with catalog number 1010311ANR for all students in the 1st semester at a (A) Bachelor - preliminary level.
We give an introduction in the algebraic structure of rings, ideals, modules and group representations. The core of the course is to deduce important conclusions from elementary assumptions. At the same time we try to show the ``art'' in the science ``algebra''.
The emphasis is on getting familiarised with abstract strucutres and on understanding the results and their proofs. Furthermore, one has to learn to prove indepently related results. The tutorials will be very crucial for the latter.
In order to acquire the necessary mathematical skills and to improve their communication skills, students will have to make two projects (homeworks). Also an introduction to the programming language TEX will be given: this is a program that allows to write mathematical texts. For the homework every student will receive some tasks: these can be excercises discussed during the tutorials as well as completely new excercises. It is expected that these are solved in full detail and correctly and that they are presented in a document made using TEX. Solutions have to be submitted by email and also in a hard copy version. The project will have to be defended during an oral presentation. It is expected that in a short presentation (maximum 15 minutes) the student will give a clear outline for the other students. The evaluation will be based on the mathematical content, the actual presentation and the use of TEX.
The fourth chapter contain some applications and extensions of the first three chapters. Depending on the available time, this chapter could be treated partially or completely.
Course Content:
Chapter 1: Representation Theory
1.1 Definition of representations
1.2 Examples of representations
1.3 Subrepresentations
1.4 Irreducible representations
1.5 Tensor product of representations
1.6 Characters of representations
1.7 The Lemma of Schur
1.8 Othogonality relations
1.9. The regular reperesentation
1.10 Class functions
1.11 Examples
Chapter 2: Rings and ideals
2.1 Definitions and Examples
2.2 Subrings
2.3 Ring homomorphisms
2.4 Ideals
2.5 Isomorphism theorems for rings
2.6 Prime ideals en maximal ideals
2.7 Maximal ideals
2.8 Noetherian rings
2.9 The Chinese remainder theorem
2.10 Fields of fractions
2.11 Principal and Euclidean rings
2.12 Unique factorsiation domains
Chapter 3: Modules
3.1 Modules and submodules
3.2 Homomorphism and quotient modules
3.3 Modules and group representations
3.4 Free Modules
3.5 Finitely generated modules
3.6 Modules and Euclidean domains
3.7 Exact rows and projective modules
Chapter 4: Applications
4.1 Fields and field extensions
4.2 The theorem of Cayley-Hamilton
4.3 The field of p-adic numbers
4.4 Lie algebras and Lie groups
Chapter 5: Exercises
Course notes are available on:
http://homepages.vub.ac.be/~efjesper/
Complementary study material:
Complementary study material
M. Artin, Algebra, Prentice Hall, London,
1991. (ISBN: 0-13-004763-5).
P.M. Cohn, Algebra, Vol. 1, John Wiley \& Sons,
London, 1974. (ISBN: 0-471-16431-3)
L. Rowen, Algebra, Groups, Rings and Fields,
A.K. Peters, Welleslley, 1994. (ISBN: 1-56881-028-8)
1. Student knows basic concepts from representation theory of groups, ring- and module theory.
2. Student is able to apply theoretical concepts from the theory on examples.
3. Student can think in function of problem solving, both individually and in group work.
4. Student can reconstruct proofs.
5. Student can prove independently related results.
6. Student can make connections with other theories, including group theory and functional analysis.
7. Student can write a mathematical text independently about solutions of exercises.
8. Student knows the sofwarepackage LaTex and uses it for communicating clearly presented solutions of excercises.
9. Student can orally clearly present the solutions of excercises.
10. Student can consult standard references.
The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.
Within the Other Exam category, the following assignments need to be completed:
The evaluation consists of two parts, each contributing 50% to the final score.
Part 1: Theoretical examination during the examination period: written preparation with written discussion of the questions.
Part 2: consisting of problems and exercises: this part consists of a written examination during the examination period (contributing 30% to the final score) and the results of written tasks/projects during the term (contributing 20% to the final score). The score of part 2 is determined by the written examen and the results of the one or more tasks performed during the term.
To succeed for this course, the average score of parts 1 and 2 must be at least 10/20. Furthermore, the score on each of the parts must be at least 7/10. If the score of one of the parts is lower than 7/20, the final score will be the average of both parts if this is less than 7/20 or 7/20 if the average is higher than 7/20.
A final score will only be awarded if the student has participated in all parts of the complete evaluation. However, a score of 10/20 or higher for a particular part of the evaluation will result in a dispensation of this part during the second try examination period and during the examination periods of the next academic year. The student can decide not to accept this dispensation. In such case, the student must mention this non-acceptance by e-mail to the responsible lecturer of this course, not later than August 15th (in case of non-acceptance for the second try examination), or not later than November 1st (in case of non-acceptance for the next academic year). Note that the non-acceptance of a dispensation is irrevocable.
This offer is part of the following study plans:
Bachelor of Mathematics and Data Science: Standaard traject (only offered in Dutch)