Arithmetic
Code: 100113 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| Mathematics | OT | 4 |
Contact
- Name:
- Francesc Xavier Xarles Ribas
- Email:
- xavier.xarles@uab.cat
Teachers
- David Olivar Lacambra
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
It is desirable to have completed all the compulsory algebra courses; concretely, students will be assumed to master the topics covered in Estructures Algebraiques and finite extension field Theory (basics on Galois Theory).
Objectives and Contextualisation
The course aims to be an introduction to arithmetic, mainly from an algebraic point of view, studying classical results of what is called algebraic number theory. It is in a way a natural extension of the courses of algebraic structures and Galois theory, but with an emphasis on the resolution of certain arithmetic problems.
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.
- Demonstrate a high capacity for abstraction.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- 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 collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- 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.
Learning Outcomes
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
- Effectively use bibliographies and electronic resources to obtain information.
- 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 collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
- 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.
- Understand in-depth demonstrations of some theorems of advanced algebra and assimilate the definition of new algebraic structures and constructions, relating them with other knowledge and deducing their properties.
- Use algebraic tools in different fields.
Content
I. Congruences and squares
- Bezout's identity and Euclid's algorithm.
- The invertible "mod n".
- Quadratic residues and Legendre's symbol.
- The law of quadratic reciprocity.
- Quadratic fields and roots of unity.
II. Algebraic Number Theory
- Rings of algebraic integers.
- Factorization of algebraic numbers
- Factorization of ideals.
- The ideal class group.
- The units of the rings of algebraic integers.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Theory sessions | 30 | 1.2 | 8, 1, 2, 7, 5, 3, 9 |
| Type: Supervised | |||
| Practical sessions | 6 | 0.24 | 8, 2, 3, 9 |
| Problem Sessions | 14 | 0.56 | 8, 1, 2, 7, 6, 3, 9 |
| Type: Autonomous | |||
| Individualized work | 30 | 1.2 | 8, 1, 2, 7, 6, 4, 5, 3, 9 |
| Solving problems. | 30 | 1.2 | 1, 2, 7, 5, 3 |
| Study theory | 37 | 1.48 | 8, 1, 7, 5, 3, 9 |
This subject has two hours of theory per week. In addition to the course notes, at certain times it will be necessary to complete the content of the class explanations with the bibliography or material provided by the teacher.
There will be sessions dedicated to solving problems. Any doubts that arise can be asked during class or during the teachers' consultation hours. The work on these problems is based on the concepts introduced in the theory class, the statements of the theorems, and their proofs.
In the seminars, a specific application will be practiced to solve certain Diophantine equations.
There will be a list of assignments, where the student can choose one, or propose one himself to do a small work on that topic.
In addition, the subject has a page on the "virtual campus" where the lists of problems, additional material and any information related to the subject will be uploaded.
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.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| An individual work on a diophantine equation with oral exposition in a video | 30% | 0 | 0 | 8, 1, 2, 7, 6, 5, 3, 9 |
| Final exam | 35% | 3 | 0.12 | 8, 1, 2, 6, 4, 9 |
| Problems to solve | 35% | 0 | 0 | 1, 2, 7, 4, 5, 3 |
There will be a list of problems to choose one individually, which must be solved and the solution submitted in Latex, which will count for 35% of the final grade.
There will be an individualized assignment to choose from a list (or one can be proposed, provided that the theory teacher accepts it) of which an oral presentation with a video of no more than 10 minutes must be made. In addition, the student must correctly evaluate the rest of the work of his/her classmates. This assignment will correspond to 30% of the course grade.
The rest of the grade (35%) will be obtained from a final exam where some problems with several sections must be solved.
Only the final exam can be retaken. It is important to note that, in the event of presenting to improve the grade, the student waives the previous grade.
Anyone with continuous assessment can opt out of submitting problems or work, by informing the theory teacher beforehand and that % would go towards the final exam for the subject.
Those who have a single assessment must submit the solution to one of the problems on the list of problems to submit (which will count for 35% of the final grade) on the day of the exam, and take the final exam for the subject (which will count for the remaining 65%).
Bibliography
Main
Pierre Samuel, Théorie algébrique des nombres, Hermann , Paris , 1971
I.N. Stewart, D.O. Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd edition, CRC Press, 2015.
Supplementary
A. Granville, Number Theory Revealed: a Masterclass. AMS, 2019.
K.Kato, N.Kurokawa, T.Saito, Number Theory 1, Fermat's Dream. Translation of Mathematical Monographs, vol. 186, 1996, AMS.
N.Koblitz, A Course in Number Theory and Cryptography, GTM114, Springer, 1994.
D. Lorenzini. An invitation to Arithmetic Geometry. Graduate Studies in Mathematics, vol 9, 1996, AMS.
J. Neukirch, Algebraic number theory, Springer-Verlag 1999.
J.-P. Serre, A Course in Arithmetic, GTM7, Springer, 1973.
J.J. Silverman, A friendly introduction to Number Theory, Pearson Modern Classics series.
W. Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer-Verlag, Berlin, 2008.
Software
The student can use SageMath, Pari or Magma (Magma is specific software for Algebra where many functions are introduced, and a simple program that lasts less than 2 minutes can be done online via the web for free).
Groups and Languages
Please note that this information is provisional until 30 November 2025. You can check it through this link. To consult the language you will need to enter the CODE of the subject.
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 1 | Catalan | second semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | second semester | afternoon |
| (TE) Theory | 1 | Catalan | second semester | morning-mixed |