Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 0 |
It is desirable to have completed all the compulsory algebra courses; concretely, students will be assumed to master the topics covered in Estructures Algebraiques.
The goal of this course is to introduce the student to arithmetic while, at the same time, offering a view of the methods that play a role in their analysis and resolution. Since there is a vast range of areas that fit inside number theory, this course will be based mainly on diophantine problems, from which algebraic number theory and arithmetic geometry will be introduced.
The course will be divided in four parts: (I) Divisibility and congruences; (II) Elliptic curves; (III) Quadratic reciprocity law; and (IV) Primality and factorization. The common theme among these, which can serve as motivation – although this is not the focus of the course –, is the applications they have found in cryptography.
In the first part we will study the basic results on prime numbers and factorization, and we will see the first applications to cryptography.
The second part will be devoted to elliptic curves, emphasizing their applications to factorization and cryptography.
In the third part we will introduce the law of quadratic reciprocity and its consequences.
In the fourth part we will investigate algorithms to determine the primality of integers, or to find nontrivial factors of composites.
Contrary to what could be thought, number theory is one of the branches of mathematics that most closely resembles experimental sciences: its main object of study is something as concrete as numbers, which we know and use in our daily lives. This is why experimentation is a fundamental trait of number theory, and this is reflected in the course by using computer tools (mainly Sage) that allow us to discover, understand and solve many arithmetic phenomena.
I. Primes and congruences
II. Elliptic curves
III. Quadratic reciprocity law
IV. Primality and factorization
This subject has two weekly hours of theory. Other than the supplied lecture notes, at some points of the course it will be useful to read additional bibliography or material provided by the teacher.
There will be sessions dedicated to solving problems. Each student will have to present one of the problems in the list resolved, in writing and delivered to the teacher. The doubts that may arise may be asked during the class or during the consultative hours of the teachers. Work on these problems is based on the concepts introduced in the theory class, the statements of the theorems, and their demonstrations, since very often the techniques will be similar.
During the seminars, we will use SAGE to solve a project.
In addition, the subject has a page in the "Virtual Campus" where the lists of problems, additional material and any information related to the subject will be uploaded.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Theory sessions | 30 | 1.2 | 1, 2, 7, 5 |
Type: Supervised | |||
Practical sessions | 6 | 0.24 | 2, 3 |
Problem Sessions | 14 | 0.56 | 1, 2, 7, 3 |
Type: Autonomous | |||
Study theory | 37 | 1.48 | 1, 5, 3 |
Work on problems and computer programming | 60 | 2.4 | 1, 2, 7, 5, 3 |
During the course the student will have to turn in a problem, worth 25% of the final grade. The student will have to code a Sage program to apply some of the techniques explained in class, among several proposals that will be made at the beginning of the course, and worth 20% of the final grade. There will also be an oral presentation worth 25%. The remainign part of the grade (30%) will be obtained from a final exam consisting of several problems.
The only second chances will be given for the final exam and/or the Sage project, as long as the grade in each part is above 3,5 / 10. It is important to underline that, in case of trying a second chance, the student gives up on the previous mark.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
Final exam | 30% | 3 | 0.12 | 8, 1, 2 |
Oral exposition | 25% | 0 | 0 | 1, 2, 7, 6, 5, 3 |
Problems to turn in | 25% | 0 | 0 | 1, 2, 7, 4, 5, 3 |
Program | 20% | 0 | 0 | 1, 2, 7, 5, 3, 9 |
Main
W. Stein, Elementary Number Theory: Primes, Congruences, and Secrets, Springer-Verlag, Berlin, 2008.
J.-P. Serre, A Course in Arithmetic, GTM7, Springer, 1973.
N.Koblitz, A Course in Number Theory and Cryptography, GTM114, Springer, 1994.
Supplementary
I.N. Stewart, D.O. Tall, Algebraic Number Theory, Chapman and Hall, 1979.
Z.I. Borevich y I.R. Shafarevich, Number Theory, Academic Press, 1966.
L.J. Mordell, Diophantine Equations, Academic Press, 1969.
J. Neukirch, Algebraic number theory, Springer-Verlag 1999.