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Arithmetic

Code: 100113
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OP 4

Contact lecturer

Name :
Xavier Xarles Ribas
Email :
xavier.xarles@uab.cat

Teaching staff

David Olivar Lacambra

Group languages

You can consult this information at the end of the 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

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.

Learning outcomes

  1. Effectively use bibliographies and electronic resources to obtain information.
  2. 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.
  3. Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
  4. 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.
  5. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  6. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  7. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  8. Use algebraic tools in different fields.
  9. 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.

Contents

I. Congruences and squares


  • 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.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Practical sessions 6 0.24 1, 3, 8, 9
Problem Sessions 14 0.56 1, 3, 5, 6, 7, 8, 9
Study theory 37 1.48 1, 4, 6, 7, 8, 9
Solving problems. 30 1.2 1, 3, 4, 6, 7
Individualized work 30 1.2 1, 2, 3, 4, 5, 6, 7, 8, 9
Theory sessions 30 1.2 1, 3, 4, 6, 7, 8, 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

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Final exam 40% 3 0.12 2, 3, 5, 7, 8, 9
Problems to solve 35% 0 0 1, 2, 3, 4, 6, 7
An individual work on a diophantine equation with oral exposition in a video 25% 0 0 1, 3, 4, 5, 6, 7, 8, 9

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 25% of the course grade.

The rest of the grade (40%) 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). It can be used also lean4 to formalize the solution of some of the problems.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan second semester morning-mixed
(PAUL) Classroom practices 1 Catalan second semester morning-mixed
(SEM) Seminars 1 Catalan second semester afternoon