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
After the study of the initial topics in field theory, the aim of the subject is the study of Galois theory which relates algebraic equations to field and group theory. As an application, it will be studied the solution of some classical geometric problems on constructions with ruler and compass: doubling the cube, trisecting an angle, squaring the circle and construction of regular polygons.
1. Field Extensions. (Lectures: 6 hours)
The degree of an extension.
Algebraic extensions.
2. Applications to Geometry. (Lectures: 4 hours)
Ruler and compass constructions: an algebraic approach.
3. Splitting fields. Algebraic Clousure. (Lectures: 7 hours)
Kronecker’s Theorem.
Existence and uniqueness of the splitting field of a polynomial.
Algebraic closure of a field.
4. Normal and separable extensions. (Lectures: 7 hours)
Multiplicity of polinomial roots. Separability.
Primitive element theorem.
Normal extensions.
5. Galois theory. (Lectures: 5 hours)
Galois extensions.
The fundamental theorem of Galois theory.
Finite fields.
6. Solvability of equations by radicals. (Lectures: 10 hours)
Solvable groups.
The Galois group of a polynomial: Galois solubility Theorem.
Solvability of quadratic, cubic and quartic equations.
Unsolvability of the quintic.
7. Applications. (Lectures: 3 hours)
The construction of regular polygons. Gauss-Wantzel’s theorem.
The fundamental theorem of algebra.
Bibliografía básica:
F. Chamizo, ¡Qué bonita es la la Teoría de Galois!
http://matematicas.uam.es/~fernando.chamizo/libreria/fich/APalgebraII04…
M.P. López, N. Rodríguez, E. Villanueva, Notas para un curso de Teoría de Galois,
https://www.usc.es/regaca/apuntes/Galois.pdf
J.S. Milne, Fields and Galois Theory,
https://www.jmilne.org/math/CourseNotes/FT.pdf
Bibliografía complementaria:
M.H. Fenrick, Introduction to the Galois Correspondence, Birkäuser, 1992.
D.J.H. Garling, A course in Galois Theory, Cambridge Univ. Press, 1986.
J.M. Howie, Fields and Galois Theory, SUMS Springer, 2006.
S. Lang, Algebra, Addison-Wesley, 1993.
I. Stewart, Galois Theory, Chapman& Hall/CRC, 2004.
J. Rotman, Galois Theory, Springer-Verlag, 1998.
To contribute to achieve the general, specific and transversal competentes listed in the USC Report on the Degree in Mathematics, in particular the following (CG3, CG4, CG5, CE4, CT1, CT2 e CT5):
- To apply the knowledge acquired and the capacity of analysis and abstraction for the formulation of problems and the search for their solutions.
- Communication of written and oral knowledge, methods, ideas and results.
- Identification of errors in incorrect reasonings.
- Use of searching tools for bibliographic resources on the course topics.
It will follow general methodological instructions contained in the USC Report on the Degree in Mathematics.
The large group lectures will basically consist of the presentation of theoretical concepts, some examples and the proof of the corresponding results (working on competences CE1, CE2, CE5 e CE6).
In the laboratory interactive classes, there will be a major implication of the student, giving priority to a more active and personalized pedagogy, and they will be devoted to solving problems, under the supervision of the teacher, which will also contribute to the acquisition of practical skills and to the illustration of the theoretical contents (CG3, CG4, CT3, CE1, CE3, CE4 e CE6).
Assigment proposals will be done (either individual and / or in group) throughout the four-month period (CG4, CG5, CT1, CT5, CE1 e CE3).
In the tutorials in small groups, a personalized follow-up of student learning and out-of-class work will be carried out (working on competences CG4, CG5, CT1, CT5, CE1 e CE3).
For the subject a support virtual course will be provided.
Bulletins with problems will be proposed in the virtual course, scheduling them in stages and always in relation to the theory.
It will follow the general assessment criteria set out in the bachelor's Degree in Mathematics at USC.
For the continuous assessment it will be taken into account both the controls that should be done in class and the assignments requested by the teacher as well as the participation in classes and tutorials corresponding to the academic year.
At least 40% of the final test will focus on the evaluation of the results presented in the lecture class; the student must be able to prove statements and answer questions about the lectures. The remainder of the final test will consist of problem solving and exercises.
Student assessment (also in the case of repeat students) will be based on a final test (E) and a continuous assessment of the work done throughout the four-month period (C). For the latter assessment it will be taken into account both the controls that should be done in class and the assignments requested by the teacher as well as the participation in classes and tutorials corresponding to the academic year.
For calculating the final grade (F) the continuous assessment (C) and the final exam (E) will be taken into account, and the final grade will be obtained by applying the formula: F = max (E, 0.3 * C +0.7 * E).
In cases of fraud in tests or examinations, the norms of the Regulations for the evaluation of student academic performance and review of grades will apply.
It will be considered as "No Presentado" the student that does not attend any of the final test in the first and second chance.
Following the guidelines set forth in the USC Report on the Degree in Mathematics, the time the student should devote to the preparation of the matter is estimated as follows:
- 58 hours of classroom work:
- Lectures sessions: 42 hours.
- Exercise sessions: 14 hours.
- Office sessions: 2 hours.
- 92 hours of personal work which includes the following activities:
- Independent study (50 hours).
- Writing reports and problem-solving work (37 hours).
- Document searchin (5 hours).
The students should have previously attended the course “Estructuras algebraicas”.
Assistance and active participation in scheduled classes and tutorials is recommended, supplemented with daily work necessary to understand the subject concepts and to carry out the activities (problems, reports) that will be periodically proposed.
CONTINGENCY PLAN:
TEACHING METHODOLOGY
SCENARIO 2:
Given that classroom teaching will coexist with virtual teaching and it is up to the Institution to define the coexistence formulas of both teaching modalities, once these are known, MS-Teams and the USC Virtual Campus will be used. The teaching will be carried out synchronously.
SCENARIO 3:
In this scenario, the teaching that will be totally virtual will be carried out synchronously using MS-Teams and the USC Virtual Campus.
In either scenario proposed bulletins problems in the virtual course scheduling them in stages in relation to the theory.
The tutoring sessions will be held electronically, we can also use e-mail to develop the sessions.
ASSESSMENT SYSTEM
The continuous evaluation combined with a final test will be evaluation criterion. The final test will be held on the date set by the Faculty of Mathematics for that purpose.
The continuous evaluation will consist of the individual resolution of tasks and exams that in cases of scenarios 2 and 3 will be proposed through the virtual course.
SCENARIO 2:
In the second scenario, the final test will preferably be face-to-face. For the final grade calculation (F) the continuous assessment (C) and the final exam (E) will be taken into account, and the final grade will be obtained by applying the formula: F = max (E, 0.3 * C +0.7 * E).
SCENARIO 3:
In the third scenario, the final test will be telematics and the final grade, both on the first opportunity and in the second, will be the sum of 40% of the note of the continuous assessmentand 60% of the note of the corresponding final test.
=======================================
In case of fraudulent authorship of exercises or tests, the “Regulations for the evaluation of the academic achivement of students and the review of grades” shall apply.
It will be considered as "No Presentado" the student that does not attend any of the final test in the first and second chance.
Leovigildo Alonso Tarrio
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813159
- leo.alonso [at] usc.es
- Category
- Professor: University Lecturer
Maria Purificacion Lopez Lopez
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813157
- mpurificacion.lopez [at] usc.es
- Category
- Professor: University Lecturer
Ana Jeremías López
Coordinador/a- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813366
- ana.jeremias [at] usc.es
- Category
- Professor: University Lecturer
Beatriz Álvarez Díaz
- Department
- Mathematics
- Area
- Algebra
- Phone
- 881813175
- beatriz.alvarez.diaz [at] rai.usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Graduation Hall |
| 12:00-13:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| Tuesday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Graduation Hall |
| 12:00-13:00 | Grupo /CLE_02 | Spanish | Classroom 06 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 12:00-13:00 | Grupo /CLE_02 | Spanish | Graduation Hall |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_03 | Spanish | Classroom 09 |
| 10:00-11:00 | Grupo /CLIL_02 | Spanish | Classroom 09 |
| 11:00-12:00 | Grupo /CLIL_06 | Galician, Spanish | Classroom 05 |
| 11:00-12:00 | Grupo /CLIL_01 | Spanish | Classroom 09 |
| 12:00-13:00 | Grupo /CLIL_04 | Galician, Spanish | Classroom 06 |
| 13:00-14:00 | Grupo /CLIL_05 | Spanish | Classroom 05 |
| 01.10.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 09 |
| 05.16.2022 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.13.2022 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |