The course focuses on a selection of topics from analytic and algebraic number theory. One of the big four items listed in the program below (THEORETICAL/ PRACTICAL CONTENT) will be selected each year, depending on the circumstances, and the course will deal with it. For the moment, the second topic, "Number Fields and Rings of Integers" has been selected.



More specifically, the aim of the course will be to understand how the "Fundamental Theorem of Arithmetic" (which states that every natural number greater than 1 can be written uniquely as a product of primes) can be extended to more general rings than the ring of ordinary integers, which are subrings of the field of complex numbers. These rings are the so-called rings of integers of number fields, that is, the finite extensions of the field of rational numbers.



We start from scratch, setting the existence and uniqueness of factorization in the ordinary integers. Next, we study the basic properties of principal and factorial domains. Then we pass to the study of the ring of integers of the a number field, Dedekind domains and the unique factorization theorem for ideals in these rings. Finally, a more detailed study of quadratic fields is made and the properties known so far for these rings are applied to the study of representations of integers by means of quadratic forms, to the resolution of Diophantine equations and to other related topics.



The classic example that serves as a model to what is studied in the course is Fermat's theorem on sums of two squares: an odd prime number is the sum of two squares of whole numbers if and only if it leaves remainder 1 when divided by 4. Of the several known proofs of this theorem, in our course it is highlighted the proof that can be deduced easily from the fact that the so-called ring of Gaussian integers is a factorial domain.



As a requirement to follow the course, a certain familiarity with the handling of congruences and with the basic concepts of the theory of commutative rings (homomorphisms, quotient rings, ideals, etc) is desirable. It is recommended, to get an idea about the topics, methods and ideas of the content of the course and the level at which they will be dealt in the class, to browse the first lessons of Stewart and Tall's book mentioned in the bibliography below.

COMPETENCIES



1. To apply the main methods for the study of arithmetic functions.

2. To relate different problems of number theory with arithmetic functions.

3. To know the problem of factorization in the rings of integers of number fields.

4. To know the basic facts about elliptic curves, the operation between its points and some of its properties and applications.

5. To know what are the main problems of additive number theory and its relations to other problems.



LEARNING OUTCOMES



1. To know how to deduce the laws of decomposition of primes in abelian extensions of the field of rational numbers.

2. To know how to apply the methods of algebraic number theory in the resolution of diophantine equations.

3. To be able to recognize problems of number theory whose solution depends on an elliptic curve.

4. To know how to calculate the rank and the torsion of the group of rational points of an elliptic curve in simple cases.

5. To know how to find estimates for different measures of algebraic numbers: means and measures of Mahler.

1. ARITHMETIC FUNCTIONS: Dirichlet products and means. Distribution of prime numbers: Theorem of Chebyshev. The Prime number theorem. Its elementary proof. Its analytical proof. Characters and Theorem of Dirichlet.



2. NUMBER FIELDS AND RINGS OF INTEGERS: Integral extensions of rings. Dedekind rings. Unique factorization of ideals. Laws of decomposition of primes.



3. ELLIPTIC CURVES: The group law on a cubic. Rational points. Torsion points. Theorem of Mordell-Weil. Computation of the rank.



4. ADDITIVE THEORY OF NUMBERS: Sums of squares. Partitions. Jacobi functions. The problem of Waring.

The theoretical content will be exposed in master classes following basic references that appear in the Bibliography. These master classes will be complemented by problem classes (classroom practices) in which students will apply the knowledge acquired in the theoretical lectures in order to solve problems. In the seminar sessions, exercises and representative examples will be considered. These will have been given to the students in advance, for them to have enough time to work out the solutions. Students must participate actively in the seminar sessions, and discussion of the solutions will be encouraged. Individual work on theory and problems might be proposed to the students, with the support of the lecturer, if needed, during the seminar sessions.

• Final Assessment System
• Tools and qualification percentages:
  • See GUIDELINES (%): 100

There will be a final writing exam. To pass the subject it will be enough to pass the writing exam and follow the activities in class. If the student decides to go to the final exam, the final mark will be the weighted average of the following activities, with the indicated weights:



20%, for other types of exercises, either individual or in groups, and written or with oral exposition, developed during the course.



80%, the final written exam (but, in any case, a minimum of four points out of 10 will be necessary to pass the subject)



In the event that the sanitary conditions prevent the realization of a face-to-face evaluation, a non-face-to-face evaluation will be activated, of which the students will be informed promptly.

There will be a writing exam. The final mark will be the weighted average of the following activities, with the indicated weights:



20%, for other types of exercises, either individual or in groups, and written or with oral exposition, developed during the course.



80%, the final written exam.



For the students not participating during the course in other types of exercise, the final mark will be that which is obtained in the written exam corresponding to this call.



In the event that the sanitary conditions prevent the realization of a face-to-face evaluation, a non-face-to-face evaluation will be activated, of which the students will be informed promptly

Basic bibliography

P. SAMUEL, Théorie algèbrique des nombres, Hermann, Paris, 1967.

I. STEWART, D. TALL, Algebraic Number Theory, Chapman&Hall, 1987.

In-depth bibliography

S. LANG, Algebraic Number Theory,1994.
R. LONG, Algebraic Number Theory, Marcel Dekker,1977.
D.A. MARCUS, Number Fields, Springer,1977.
T. ONO, An Introduction to Algebraic Number Theory, Plenum,1990.