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: Second Semester
Teaching: With teaching
Enrolment: Enrollable
To know some of the most important applications of mathematics to number theory and geometry.
To know the theory of quadratic residues and the law of quadratic reciprocity and the importance of the latter as a source of ideas for the theory of numbers.
To understand the meaning of fundamental theorem of arithmetic in the context of rings of algebraic integers.
To know Hilbert's Nullstellensatz and the relationship between ideals and varieties contained in the "algebra-geometry dictionary".
To appreciate the fundamental aspects of the theory of plane algebraic curves, including an introduction to intersection theory.
1. Quadratic residues. The law of quadratic reciprocity. Computation of the symbols of Legendre and Jacobi. Applications. (3 expository hours)
2. Representations of integers by binary quadratic forms. Sums of squares. Theorems of Lagrange, Euler and Legendre. (3 expository hours)
3. Algebraic number fields. The discriminant. Algebraic integers and integral bases. Quadratic and cyclotomic fields. (4 expository hours)
4. Factorization in rings of algebraic integers. The fundamental theorem of arithmetic for ideals. (3 expository hours)
5. Algebraic sets. Hilbert basis theorem. Gröbner bases. Radical ideals. Hilbert's Nullstellensatz and the algebra-geometry dictionary. The Zariski topology. (8 expository hours)
6. Projective algebraic curves. Rational applications. Smooth points and singular points. (6 expository hours)
7. Plane curves. Multiplicities and intersection numbers. Bezout's theorem. (15 expository hours)
Bibliografía básica:
Adams-Goldstein, Introduction to Number Theory, Prentice Hall 1976.
W. Fulton, Algebraic Curves. An Introduction to Algebraic Geometry, 2008. http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
Bibliografía complementaria
J.S. Chahal, Topics in Number Theory, Plenum, 1988.
W. Fulton, Introduction to intersection theory in algebraic geometry, American Mathematical Society, Providence, RI, 1984.
J.-P. Serre, Cours d'arithmétique, Presses Universitaires de France, Paris, 1977.
W. Kunz, Introduction to Plane Algebraic Curves, Birkhäuser, 2005
To contribute to achieving the generic, specific and transversal competences listed in the "Report on the Degree in Mathematics" from the USC and, in particular, the following:
Written and oral communication of knowledge, methods and general ideas related to number theory and geometry (CG4).
Use of searching tools for bibliographic resources on the course topics, including Internet access. Use of these resources in different languages, especially English (CT1, CT5).
To develop hypotheses and draw appropriate conclusions using well-reasoned arguments while identifying logical flaws and fallacies in argumentation (CG2, CE4).
Specific competences for this course:
To become familiar with the Legendre and Jacobi symbols and their efficient computation, as well as some of their main applications.
To know some of the most important classical results on representation of integers by quadratic forms and, in particular, as sums of squares.
To know the basic theory of factorization of algebraic integers and to study the problem of non uniqueness of factorization.
To study the factorization problem in the broader context of the theory of ideals and to understand the unique factorization theorem in this context.
To handle with ease the algebra-geometry dictionary. To describe basic operations in geometry and describe their counterparts in algebra.
To know some of the most important aspects of the theory of plane algebraic curves and to understand Bezout's theorem.
Scenario 1: The general methodological indications that appear in the USC Mathematics Degree Title Report will be followed. Teaching will be taught in blackboard classes and tutorials.
In the classes, the essential contents of the subject will be presented, problems will be solved and the activities that the students must carry out for the continuous evaluation will be proposed: problem solving, work preparation, ... (competences CB3, CB4, CG1, CG2, CG4, CG5, CT1, CT5).
Scenario2: classroom teaching will be in blackboard classes and tutorials. The virtual one will be in asynchronous mode. The platforms will be those provided by the academic authorities.
Scenario 3: teaching will be in asynchronous mode. The platforms will be those provided by the academic authorities.
Scenario 1: Continuous evaluation: two partial tests in the exposition hour
For the calculation of the final grade (F) the continuous assessment (C) and the final exam grade (E) will be taken into account and the following formula will be applied:
F = max (E, C / 3 + 2E / 3).
All this for both opportunities
Scenario 2: Continuous evaluation: a partial test in the exposition hour. Delivery, via e-mail, within a specified period, of several exercises, which will be uploaded to the Virtual Classroom, common for all students, but with optional. The grades will be communicated in Virtual Classroom, before the final exam, without reviewing them.
Final telematic exam: 2 exercises, on the day and time indicated by the Center. It will be weighted with the following formula:
F = max (E, c + e / 4),
E = final test mark or 0, c = C if Cmenor or equal5, 5+ (C-5) / 2 if Cmayor5, C = continuous evaluation, e = 0 if Emenor or equal5, = E if Emayor5.
All this for both opportunities.
Scenario 3: Continuous assessment: Delivery, via e-mail, within a period to be indicated, of several exercises, which I will upload to the Virtual Classroom, common to all students, but optional. The grades will be communicated in Virtual Classroom, before the final exam, without reviewing them.
Final telematic exam: 2 exercises, on the day and time indicated by the Center. It will be weighted with the following formula:
F = max (E, c + e / 4),
E = final test mark or 0, c = C if Cmenor or equal5, 5+ (C-5) / 2 if Cmayor5, C = continuous evaluation, e = 0 if Emenor or equal5, = E if Emayor5.
All this for both opportunities.
In cases of fraudulent performance of exercises or tests, the provisions of the Academic Performance Assessment Regulations will apply.
Following the guidelines set forth in the Report on the Degree in Mathematics from the USC, the time the student should devote to the preparation of the matter is:
- 58 hours of classroom work.
- 92 hours of personal work which includes the following activities:
- 52 hours of independent study.
- Writing reports and problem-solving work: 25 hours.
- Recommended reading and document searching: 5 hours.
- Preparation of oral presentations: 10 hours.
Prior knowledge of the basic algebraic structures, including field extensions, is advisable.
Attendance and active participation in scheduled classes and tutorials is recommended, supplemented with daily work necessary to assimilate the subject concepts and to carry out the activities (problems, reports) that will be periodically proposed.
Contingency Plan
Teaching methodology
Scenario2: classroom teaching will be in blackboard classes and tutorials. The virtual one will be in asynchronous mode. The platforms will be those provided by the academic authorities.
Scenario 3: teaching will be in asynchronous mode. The platforms will be those provided by the academic authorities.
Evaluation system
Scenario 2: Continuous evaluation: a partial test in the exposition hour. Delivery, via e-mail, within a specified period, of various exercises, which I will upload to the Virtual Classroom, common for all students, but with optional. The grades will be communicated in Virtual Classroom, before the final exam, without reviewing them.
Final telematic exam: 2 exercises, on the day and time indicated by the Center. It will be weighted with the following formula:
F = max (E, c + e / 4),
E = final test mark or 0, c = C if Cmenor or equal5, 5+ (C-5) / 2 if Cmayor5, C = continuous evaluation, e = 0 if Emenor or equal5, = E if Emayor5.
All this for both opportunities.
Scenario 3: Continuous assessment: Delivery, via e-mail, within a period to be indicated, of several exercises, which I will upload to the Virtual Classroom, common to all students, but optional. The grades will be communicated in Virtual Classroom, before the final exam, without reviewing them.
Final telematic exam: 2 exercises, on the day and time indicated by the Center. It will be weighted with the following formula:
F = max (E, c + e / 4),
E = final test mark or 0, c = C if Cmenor or equal5, 5+ (C-5) / 2 if Cmayor5, C = continuous evaluation, e = 0 if Emenor or equal5, = E if Emayor5.
All this for both opportunities.
Leoncio Franco Fernández
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813163
- leoncio.franco [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 17:00-18:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Tuesday | |||
| 17:00-18:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 19:00-20:00 | Grupo /CLIL_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Wednesday | |||
| 16:00-17:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 06.07.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 02 |
| 06.07.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 03 |
| 07.05.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |