GALOIS THEORY
Academic year and teacher
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- FABIO STUMBO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/02
Training objectives
- The goal of the course is to develop the theory of fields necessary to
understanding of Galois correspondence and its main consequences.
Main acquired knowledges are:
* Structure of finite fields;
* Resultant of two polynomials;
* Properties of field extensions (algebraicity, separability, normality, simplicity, transcendence);
* Constructibility of algebraic numbers;
* Solvability by radicals of an equation;
* Galois correspondence;
* Criteria for the determination of the Galois group of a polynomial;
* Cyclotomic extensions.
Main acquired skills are:
* Knowing how to calculate the resultant of two polynomials;
* Knowing how to determine a generator of simple extension;
* Be able to recognize the properties of an extension of fields;
* Knowing how to make constructions with ruler and compass;
* Calculation of the Galois group of a polynomial. Prerequisites
- Prerequisites:
Elementary algebra knowledge as acquired in a first year undergraduate
Algebra course. In particular:
* Notions about groups, rings and fields;
* Structure of finite fields;
* Calculation of the GCD and the Euclidean algorithm. Course programme
- 1. Field theory (12 hours).
* Separable extensions.
* Normal extensions.
* Splitting fields.
* Primitive elements.
* Algebraic closure.
* Fundamental theorem of algebra.
2. Galois thoery (12 hours).
* Galois correspondance.
* Galois extensions.
* Fundamental theorem.
3. Galois theory of equations (12 hours).
* Solvability by radicals.
* Solvable groups.
* Solvable polynomials.
* Ruffini-Abel's theorem.
* Constructions with ruler and compass.
4. Additional topics (12 hours);
* Transcendence;
* Transcendental extensions;
* Galois' original work. Didactic methods
- The course is organized in conventional lectures, usually 2 hours each.
Lectures will be held in the classroom and/or streaming, as needed. Learning assessment procedures
- Examination test consists in verifying the achievement of expected skills.
This verification is through a written test lasting 3 hours and consisting in exercises related to the topics of the course lectures.
The written test consists of questions and/or exercises with variable score for a total of not less than 30.
The total number of points obtained in the test (limited to 30 if greater) is the student's grade.
The examination is passed if the grade is at least 18.
Oral session is optional, at student's request.
Any oral session contributes to the final mark with a score between -3 and 3. Reference texts
- Reference books:
* F. Stumbo: Teoria di Galois, course notes
Additional references:
* C. Menini, F. Van Oystaeyen; Abstract algebra; Marcel Dekker
* M. Girardi, G. Israel; Teoria dei campi; Feltrinelli
* D. A. Cox; Galois theory; Wiley
* H. M. Edwards; Galois theory; Springer
* J. M. Howie; Fields and Galois theory; Springer
* S. Roman; Fields theory; Springer
* J. Rotman; Galois theory; Springer
