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Galois Theory

Code: 100102
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OB 3

Contact lecturer

Name :
Ramon Antoine Riolobos
Email :
ramon.antoine@uab.cat

Teaching staff

Jaume Coll Guerrero

Group languages

You can consult this information at the end of the document.

Prerequisites

Background on Group Theory is required (e.g. the notions introduced and studied in "Estructures Algebraiques"). Groups are used in an essential way in this module. Thus, in order to be able to work with concrete examples, it is particularly interesting to have some familiarity with groups of small order and Sylow Thorems.

It is also important to be familiar with basic notions of Ring Theory (again, the ones introduced in "Estructures Algebraiques). Of particular importance are the notions related to irreducible polynomials, as well as the construction of fields as quotients of polynomial rings.

Objectives

The main objective of this module is to develop the notions of Galois Theory and their applications to problems related to resolubility of equations. The latter problems arise as some of the oldest in the History of Mathemathics. Their roots can be traced back to the Babylonia era and culminates brilliantly with the work of Évarist Galois, whose work develops the theory of solvability by radicals.

The modern approach to Galois Theory constitutes a central theme in Algebra, since the abstract methods used show the power of (previously introducted) tools in action. Thus, the translation of a problem to Field Theory, and subsequently to Group Theory (and back) show how abstract, seemingly different branches of of Mathematics interact to solve a classical, more applied problem.

We will start introducing the problem of solving an equation by radicals in its historical context. Next, Field Theory will provide the formal framework where to formulate the problem and study effectively the Galois Theory of equations.

A fundamental tool here is provided by the techniques coming from Group Theory, particularly when it comes to examples and manipulation. However, due to time constraints, we shall review only the most basic concepts and refer to the notions studied in the course "Estructures Algebraiques".

 

Learning outcomes

  1. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  2. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  3. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  4. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  5. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  6. Calculate the maximum common divisor and factorisation of whole numbers and polynomials.
  7. Relate geometric constructions with algebraic extensions.
  8. Operate in some simple groups (such as cyclic, dihedral, symmetric and abelian).
  9. Construct quotient groups and rings and finite bodies and operate within them.
  10. Calculate groups of low degree Galois equations and deduce their resolvability by radicals.
  11. Manipulate expressions involving algebraic and transcendent elements.

Contents

1. Solvability of equactions and ring plreliminaires


2. Field extensions


3. Normal and separable extensions


4. The Fundamental Theorem of finite Galois Theory


5. Galois theory of equations.


 

Learning activities and methodology

Title Hours ECTS Learning outcomes
Problem solving 40 1.6 2, 4, 7, 9, 10, 11
Exams preparation 16 0.64 2, 4, 7, 10, 11
Course work (from lectures) 27 1.08 1, 2, 4, 7, 10, 11
Seminars 6 0.24 1, 2, 3, 4, 5, 7, 9, 10, 11
Seminar preparation 10 0.4 2, 3, 4, 7, 9, 10, 11
Lectures 30 1.2 1, 2, 3, 4, 5, 7, 9, 10, 11
Tutorials 15 0.6 1, 2, 3, 4, 5, 7, 9, 10, 11

There will be two lectures and one tutorial per week, during 15 weeks. In addition, there will be 3 seminar sessions of 2 hours each, distributed in the semester. Students are strongly encouraged to attend lectures, tutorials, and seminars.

During the lectures, the main tools needed for understanding the subject and also for problem-solving will be introduced.

Problem-solving will be the main focus in the tutorials, where also a better understanding of the concepts introduced in the lectures will be achieved. Students participation in the form of discussion will be part of the methodology.

In seminars, students participation will be more prominent as these are designed in the form of hands-on exercises and focusing, in particular, in manipulation of examples.

Various resources will be offered through moodle. In particular, problems/seminars and additional material that may complement the subject of the course.

 

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Seminars 15% 1 0.04 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Exam 50% 3 0.12 1, 2, 4, 5, 6, 7, 8, 9, 10, 11
Intersemester exam 35% 2 0.08 1, 2, 4, 5, 6, 7, 8, 9, 10, 11

The subject will be evaluated as follows:

  • 35% of the grade will correspond to the completion of a partial exam.
  • 15% of the grade will correspond to seminar evaluation.
  • 50% of the grade will correspond to the completion of a final exam.


In the case of a single assessment, there will be a final exam corresponding to 100% of the final grade that will be held to coincide with the date of the final exam.

There will be a second chance exam, both for the continuous assessment and for the single assessment, which will make it possible to recover the grade of the exams in the event that the average of the subject is lower than 5.

The grade of non-evaluable will be obtained only if neither the final exam nor the retake is taken.

 

 

Bibliography

David A. Cox, "Galois Theory". Hoboken : Wiley-Interscience, cop. 2004

Harold M. Edwards, "Galois Theory". Springer-Verlag GTM 101, 1993

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

J. Milne, "Fields and Galois Theory". http://www.jmilne.org/math/

Ian Stewart, "Galois Theory". Chapman & Hall CRC, 2004

Jean-Perre Tignol, "Galois' Theory of Algebraic Equations". World Scientific 2001



Software

We will use SageMath. 

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan first semester morning-mixed
(SEM) Seminars 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 2 Catalan first semester morning-mixed
(SEM) Seminars 2 Catalan first semester morning-mixed