Algebraic Equations

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 Closure. (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 polynomial roots. Separability.
Finite fields.
Primitive element theorem.
Normal extensions.

5. Galois theory. (Lectures: 5 hours)
Galois extensions.
The fundamental theorem of Galois theory.
Cyclotomic extensions.

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.

Basic bibliography:

F. Chamizo: ¡Qué bonita es la la Teoría de Galois!
http://matematicas.uam.es/~fernando.chamizo/libreria/fich/APalgebraII04…

D. A. Cox: Galois theory. 2ª ed., John Wiley & Sons, NJ, 2012.

T. Leinster, Galois Theory, University of Edinburgh, 2023.
https://www.maths.ed.ac.uk/~tl/gt/gt.pdf

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



Complementary bibliography:

D. S. Dummit, R. M. Foote: Abstract algebra. 2ª ed., John Wiley & Sons, NJ, 2004.

M. H. Fenrick: Introduction to the Galois Correspondence, Birkäuser, 1992.

J. B. Fraleigh: A first course in abstract algebra (historical notes by Victor Katz). 8ª ed., Person, 2021.

D. J. H. Garling: A course in Galois Theory, Cambridge Univ. Press, Cambridge, 1986.

J. M. Howie: Fields and Galois Theory, SUMS Springer, London, 2006.

J. Rotman: Galois Theory, Springer-Verlag, NJ, 1998.

I. Stewart: Galois Theory, 4ª ed., CRC Press, Boca Raton, FL, 2015.

To contribute to achieve the general, specific and transversal competencies 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 lecture classes will basically consist of the presentation of theoretical concepts, some examples and the proof of the corresponding results (working on competencies 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).

Assignment proposals will be done related with the subject (CG4, CG5, CT1, CT5, CE1 e CE3).

In the tutorials in small groups, a personalized follow-up of students learning and out-of-class work will be carried out (working on competencies 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.

The assessment system will be the coordinated for the two groups of the subject.
As assessment system, it will be considered continuous assessment combined with a final exam. This final exam will be celebrated on the date fixed by the Faculty of Mathematics. The final exam will be the same for all the students of the subject.
The continuous evaluation will consist of the individual resolution of assignments (one or two in the course) and tests (one or two in the course), which will be the same for both groups.

COMPUTATION OF FINAL QUALIFICATION
The final test will be presential, both in the first and the second opportunity. At least 40% of the final test will focus on the evaluation of the theoretical 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.

On both opportunities, 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=máx(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 "Not Presented" the student that does not attend any of the final tests 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:
- Lecture classes: 42 hours.
- Interactive laboratory classes: 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).
- Documentation search and recommended readings (5 hours).

The students should have previously attended the course “Algebraic structures”.
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.