Algebraic structures
Code: 100096 ECTS Credits: 9| Degree | Type | Year | Semester |
|---|---|---|---|
| 2500149 Mathematics | OB | 2 | 2 |
Contact
- Name:
- Dolors Herbera Espinal
- Email:
- Dolors.Herbera@uab.cat
Use of Languages
- Principal working language:
- catalan (cat)
- Some groups entirely in English:
- No
- Some groups entirely in Catalan:
- Yes
- Some groups entirely in Spanish:
- No
Teachers
- Francesc Xavier Xarles Ribas
Prerequisites
The previous academic requirements will be found in the subjects Fundamentals of Mathematics and Linear Algebra, first year.
The skill acquired in algebraic manipulations, and the familiarity with operations in arithmetic contexts or groups of permutations, will continue to be developed, moving to a higher level of abstraction, which is very common in Mathematics. References to vector spaces as a model of algebraic structure and to your knowledge of matrix manipulation will also be frequent. Matrices will be a particularly important source of examples.
Objectives and Contextualisation
The objectives of this subject are of two types: to achieve
training in basic algebra and gaining knowledge and skills
to manipulate abstract objects.
Among the training objectives we highlight the following:
correctly understand and use language and mathematical reasoning in general and algebraic reasoning in particular. Be able to make small demonstrations, develop meaning
critical of mathematical statements,
develop combative attitudes and creativity in the face of problems and, finally, learn to apply abstract concepts and results in concrete examples. Present reasoning or a problem in public and develop agility to answer mathematical questions in a conversation.
Competences
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Identify the essential ideas of the demonstrations of certain basic theorems and know how to adapt them to obtain other results.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Understand and use mathematical language.
Learning Outcomes
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
- Calculate the maximum common divisor and factorisation of whole numbers and polynomials.
- Construct quotient groups and rings and finite bodies and operate within them.
- Operate in some simple groups (such as cyclic, dihedral, symmetric and abelian).
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
Content
The subject is organized in four parts:
I. Group Theory.
- Groups, subgroups and morphisms. Basic examples.
- Lateral classes. Lagrange Theorem.
- Normal subgroups, quotient group.
- Isomorphism theorems.
- Classification of cyclic groups. More on abelian groups.
- Action of a group on a set.
- Sylow's theorems.
II. Commutative rings
- Rings, ideals and morphisms. Basic examples
- Quotients and isomorphism theorems.
- Maximal and prime ideals. Zorn's Lemma.
- Field of fractions of a domain.
- The ring of polynomials
III. Factorization.
- Domains of main ideals.
- Unique factorization domains.
- Gaussian lemma. Factorization in rings of polynomials.
IV. Finite fields.
- Fields, subfieds and characteristic of a field.
- Primitive element theorem for finite fields.
- Existence and uniqueness of finite fields.
- Frobenius morphism.
Methodology
This subject has three ours per week of theory lasses, one hour per wek of problem classes, and, during the semester, eight seminar sessions, two ours each.
Students will have the lists of problems previously to be able to work before the problem classes. In classe, you can not solve all the problems but we recommend that students work on their own and asj the teachers their questions. In the seminar sessions the students will work under the supervison of the teacher. In some of these seminars, some exercises will be given that will count for the final mark of the subject.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Directed | 16 | 0.64 | |
| Theory classes | 43 | 1.72 | |
| Type: Supervised | |||
| Sminars | 14 | 0.56 | |
| Type: Autonomous | |||
| Seminar preparation | 145 | 5.8 |
Assessment
20% of the mark of the course corresponds to the delivery of problems to the seminars (S).
There will be a written test, in the middle of the semester, to evaluate the theoretical and practical abilities of the subject. The date of the test will be set by the degree coordinator. The mark on 10 (P) of this test will correspond to 30% of the total mark.
50% of the mark corresponds to that obtained in the final exam (F). In this exam
the theoretical and practical knowledge of the subject will be evaluated.
If in F a mark greater than or equal to 3.5 is obtained, then the student obtains the mark N = 0.20 · S + 0.30 · P + 0.50 · F. The subject will be passed, therefore, if the grade N is equal to or higher than 5 and if, at least, a 3.5 has been taken in the final exam.
Honor registrations will be awarded based on the value of the mark N.
There will be a recuperation exam corresponding to the final exam. Only students who have obtained a grade of less than 3.5 in the final exam or who have a grade of N <5 will be able to take this exam. In this case, we will calculate the value N '= MAX (N; 0.20 · S + 0.30 · P + 0.50 · R), where R denotes the mark of the recuperation exam and the final mark will be N' provided this value does not exceed 7, otherwise, the final mark will be 7.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Autonomous | 145 hours , 5,8 ECTs | 4 | 0.16 | 2, 3, 1, 4, 8, 7, 6, 5 |
| Supervised | 14 hours, 0,56 ECTS | 3 | 0.12 | 2, 3, 1, 4, 8, 7, 6, 5 |
Bibliography
[1] R. Antoine, R. Camps, J. Moncasi. Introducció a l'àlgebra abstracta. Manuals de la UAB, Servei de Publicacions de la UAB, no. 46, Bellaterra, 2007.
[2] F. Cedó, V. Gisin, Àlgebra bàsica, Manuals de la UAB, Servei de Publicacions de la UAB, no. 21, Bellaterra, 2007.
[3] David S. Dummit and Richard M. Foote, Abstract Algebra, 3rd. Edition, Wiley, 2003.
[4] J.B. Fraleigh. A First course in abstract algebra. Pearson Education, 7th Edition, 2014. Review: https://www.maa.org/press/maa-reviews/abstract-algebra
[5] T. W. Hungerford, Abstract Algebra, Brooks/Cole, 2013. Review:
https://www.maa.org/press/maa-reviews/abstract-algebra-an-introduction
Software
It is not planned to use any specific software in the subject. However, an algebraic manipulator (Maple, Sage, ....) can be useful when making calculations.
There are specific programs for manipulating groups such as GAP - Groups, Algorithms, Programming -
a System for Computational Discrete Algebra that is useful to know and that can solve most of the most computational problems of groups.