ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Algebra
Center Faculty of Mathematics
Call:
Teaching: Sin docencia (Extinguida)
Enrolment: No Matriculable
To become familiar with some of the most important applications of mathematics to number theory and algebraic geometry.
To study number fields and their rings of integers, and to understand the usefulness of working with these structures to address certain classical problems, such as solving Diophantine equations.
To become familiar with the basic concepts of algebraic geometry as well as some fundamental results in the aforementioned theory.
The course consists of two clearly distinct parts: one focused on number theory and another on algebraic geometry. Additionally, topics from commutative algebra will be addressed transversally, serving as a common tool for the study of both number theory and algebraic geometry. The first block contains 5 topics, and the second block contains 4.
1. Review of Classical Arithmetic (4 lecture hours). Pythagorean triples and the congruent number problem; the ring of Gaussian integers and applications.
2. Rings of Integers (3 lecture hours). Quadratic extensions of the rationals; the ring of integers of a number field; norm, trace, and discriminant.
3. Dedekind’s Theorem (3 lecture hours). The problem of unique factorization with examples; Dedekind domains and their properties; Dedekind’s theorem.
4. The Class Group and the Unit Group (4 lecture hours). Ideals and quadratic forms; Minkowski’s theorem and applications; the class group; Dirichlet’s theorem and applications.
5. Prime Factorization in Extensions (7 lecture hours). Ramification index and inertia degree; decomposition and inertia subgroups; the Frobenius element. Introduction to local fields.
6. The Correspondence between Algebra and Geometry (5 lecture hours). Affine algebraic sets; Hilbert’s basis theorem; Hilbert’s Nullstellensatz.
7. Affine and Projective Space (5 lecture hours). Affine space: coordinate rings and polynomial maps; projective space and projective linear varieties; affine vs. projective varieties.
8. Affine Curves (5 lecture hours). Multiple points and tangent lines; multiplicities and local rings; intersection numbers.
9. Projective Curves (6 lecture hours). Bézout’s theorem; Max Noether’s theorem; applications.
Basic references:
Ireland, K.; Rosen, M., A Classical Introduction to Modern Number Theory.
Springer-Verlag, 1990.
Marcus, D. A., Number Fields.
Springer-Verlag, 1977.
Fulton, W., Curvas algebraicas.
Ed. Reverté, 1971.
Complementary references:
Fernando, J. F., Curvas Algebraicas.
Ed. Sanz y Torres, 2022.
Hartshorne, H., Algebraic Geometry.
Springer-Verlag, 1977.
Kirwan, F., Complex Algebraic Curves.
Cambridge University Press, 1992.
Neukirch, J., Algebraic Number Theory.
Springer-Verlag, 1977.
Swinnerton-Dyer, H. P. F., A Brief Guide to Algebraic Number Theory.
London Math. Soc., 2001.
To contribute to achieving the general, specific, and transversal competences outlined in the Degree Programme in Mathematics at the University of Santiago de Compostela (USC), and in particular, the following:
Written and oral communication of knowledge, methods, and general ideas related to number theory and geometry (CG4).
Ability to present hypotheses and draw conclusions using well-reasoned arguments, while being able to identify logical flaws and fallacies in reasoning (CG2, CE4).
Specific competences of the subject:
To understand the basic theory of ideal factorization in the context of rings of algebraic integers.
To apply this knowledge to the resolution of classical problems such as sums of squares or certain cases of Fermat’s Last Theorem.
To become familiar with the Legendre and Jacobi symbols and their efficient computation, as well as some of their main applications.
To handle comfortably the algebra-geometry dictionary. To describe basic operations in geometry and identify their counterparts in algebra.
To understand the most important aspects of the theory of plane algebraic curves and to grasp Bézout’s Theorem.
The general methodological guidelines set out in the Degree Programme in Mathematics at USC will be followed.
Teaching will be delivered through chalkboard lectures and tutorials.
Students may present certain aspects of the subject during interactive classes.
Throughout the semester, three different grades will be obtained and used to calculate the continuous assessment grade (C):
- One grade will correspond to the work on problem sets (E). Throughout the course, students will be required to submit some exercises from the problem lists, which may be done individually or in pairs. These will always be routine exercises aimed at reinforcing the content covered in lectures.
- Another grade will correspond to short quizzes during problem-solving classes (P), which will consist of solving an exercise from the problem set. These quizzes may also be done individually or in pairs.
- A third grade will correspond to a small project (T), to be carried out individually or in pairs. This project must be presented during one of the tutorial sessions in very small groups. The evaluation of the project will consist of two parts: a written report and an oral presentation. The final grade will be the arithmetic mean of these two components.
The continuous assessment grade will be calculated as C = 0.3E + 0.3P + 0.4T. Active participation in class, by answering questions or solving problems on the board, may add up to one extra point in the continuous assessment. The final course grade will also take into account the final exam (F), using the formula min{10, max{F, 0.8C + 0.3F}}.
These specifications will also apply to the resit (second) exam session.
In accordance with the guidelines established in the Degree Programme in Mathematics at USC, the time students are expected to dedicate to the preparation of the course consists of:
58 hours of face-to-face instruction.
92 hours of individual work, including the following activities:
52 hours of independent study.
Preparation of assignments and problem solving: 25 hours.
Recommended readings and research for documentation: 5 hours.
Preparation of oral presentations: 10 hours.
Attend classes regularly.
Complete the exercises proposed by the instructor and keep up with the course material.
Seek help during office hours when difficulties arise.
Oscar Rivero Salgado
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- oscar.rivero [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
| Tuesday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLIL_01 | Galician | Classroom 09 |
| 13:00-14:00 | Grupo /CLE_01 | Galician | Classroom 09 |
| 06.01.2026 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 06.29.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |