The main goal of this course is the study of finite Galois field extensions in order to know the concept of Galois group of a polynomial, how to calculate it in simple cases, and to understand the relation of this group with the solvability by radicals of the polynomial. Before that, we introduce the basic theory of fields, algebraic extensions of fields and the splitting field of a polynomial over a field.



A level of B2 or higher is recommended to attend courses taught in English.This course belongs to the module Algebraic Structures (2nd year) + Commutative Algebra (3rd year) + Algebraic Equations (3rd year), which is devoted to developing the fundamentals of abstract algebra and its main applications. The student will learn the basic techniques in this area that will allow him to use these concepts in other areas of mathematics, as well as to embark on a deeper study of algebra in the optional courses of the 4th year, if he/she wishes to do so.

SPECIFIC COMPETENCES:

M01CM10: To know how to operate in easy field extensions.

M01CM11: To know the concepts of normal and Galois field extensions and to know how to calculate the Galois group of easy Galois extensions.

M01CM12: To know how to apply the fundamental theorem of Galois theory in order to calculate the intermediate fields of easy Galois extensions.

M01Cm13: To know how to characterize the algebraic equations which are soluble by radicals.



LEARNING RESULTS:

To know the Galois group of a polynomial and how to calculate it in easy cases. To understand the relation of this group with the solvability of a polynomial by radicals.

1. THE PROBLEM OF THE SOLVABILITY OF ALGEBRAIC EQUATIONS: What is to solve an algebraic equation? Solvability by radicals of the equations of degree at most 4. Review of polynomial rings: divisibility and irreducibility criteria. Fields, generalities. Structure of the additive and the multiplicative group of a field. Characteristic of a field and prime subfield.



2. FIELD EXTENSIONS: Field extensions. Algebraic and transcendental elements. Simple extensions, algebraic extensions, and finite extensions. Splitting field of a polynomial: existence and unicity.



3. NORMAL EXTENSIONS AND SEPARABLE EXTENSIONS: Normal extensions. Characterization of finite normal extensions. Finite separable extensions: the primitive element theorem.



4. GALOIS EXTENSIONS: Field automorphisms. Galois extensions and the Galois group. The fundamental theorem of Galois theory. Applications (finite fields, the Fundamental Theorem of Algebra).



5. SOLVABILITY OF ALGEBRAIC EQUATIONS: Solvable groups. Galois' theorem on the solvability of algebraic equations by radicals.

The theoretical contents will be presented in master classes following basic references in the bibliography. These lectures will be complemented with problem classes (classroom practice), 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 give 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.

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

There will be two written exams, one after two thirds of the course have been covered, and another one at the end of the course. The final mark will be the weighted average of the following activities, with the indicated weights:



- 50-80%, the final exam, which could be fully a written exam or a written exam for the exercises and an oral test for the theory.

- 20-50%, the partial written exam, other types of exercises, either individual or in groups, and written or with oral exposition.



The interest and willingness of the student will also be taken into account. In order to pass the course, it is necessary to obtain at least 4,5 points out of 10 in the final written exam.



The final evaluation will consist of an exam of the entire subject. 100% weight.

The final mark will be that which is obtained in the written exam corresponding to this call.

Basic bibliography

1.- CLARK, A. Elementos de Algebra Abstracta. Alhambra, Madrid, 1979.

2.- De VIOLA-PRIOLI. A.M.; VIOLA-PRIOLI, J.E. Teoría de Cuerpos y Teoría de Galois. Reverté, Barcelona, 2006.

3.- NAVARRO, G. Un curso de Algebra. Universidad de Valencia, 2002.

4.- STEWART, I. Galois Theory. Chapman & Hall, 2nd ed., London, 1989.

5.- VERA LÓPEZ, A. Introducción al Algebra, II. Ellacuría, Bilbao, 1986.

6.- VERA, A.; VERA, J. Problemas de Algebra, I: Teorías de Grupos y de Cuerpos. AVL, 1995.

In-depth bibliography

1.-GARLING, D. J. H. A course in Galois Theory. Cambridge University Press, Cambridge, 1986.
2.-HUNGERFORD, T.W. Algebra. Springer-Verlag, New York, 1984.
3.-LANG, S. Algebra. 3rd. ed. Springer, 2005.
4.-MORANDI, P. Field and Galois Theory, Springer, New York, 1996.
5.-VERA, A.; ARREGI, J.M. Problemas de Algebra, II: Teorías de Grupos, Cuerpos y Anillos. AVL, 1989.

Web addresses

http://www-history.mcs.st-andrews.ac.uk/Biographies/Galois.html
http://mathworld.wolfram.com/topics/AlgebraicEquations.html