M7230 Galois Theory

Faculty of Science
Autumn 2024
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer)
Mgr. Pavel Francírek, Ph.D. (seminar tutor)
Mgr. Jan Vondruška (assistant)
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 10:00–11:50 M5,01013
  • Timetable of Seminar Groups:
M7230/01: Wed 12:00–13:50 M6,01011, P. Francírek
Prerequisites
M3150 Algebra II
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
In the course M3150 Algebra II, we have learned the rudiments of Galois theory of finite extensions, including the main theorem of Galois theory. After recalling what we already know, we shall study Galois theory of finite extensions in full details, including some of its applications in algebra and geometry. Then we introduce topological groups and profinite groups, to be able to study Galois theory of infinite extensions.
Learning outcomes
At the end of this course, students should be able to:
apply main results on Galois theory to concrete exercises;
explain basic notions and relations among them.
Syllabus
  • Basic theory of field extensions: simple algebraic extension, the degree of extension, algebraic and transcendental extension.
  • Classical straightedge and compass constructions.
  • Splitting fields and algebraic closures.
  • Separable and inseparable extensions.
  • Cyclotomic polynomials and cyclotomic extensions.
  • Basic definitions of Galois theory.
  • The fundamental theorem of Galois theory.
  • Composite extensions and simple extensions.
  • Cyclotomic extensions and Abelian extensions over Q.
  • Galois groups of polynomials.
  • Solvable and simple groups.
  • Solvable and radical extensions: insolvability of the quintic.
  • Topological groups.
  • Infinite Galois theory.
Literature
  • DUMMIT, David Steven and Richard M. FOOTE. Abstract algebra. 3rd ed. Hoboken, N.J.: John Wiley & Sons, 2004, xii, 932. ISBN 0471433349. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
  • RAMAKRISHNAN, Dinakar and Robert J. VALENZA. Fourier analysis on number fields. New York: Springer-Verlag, 1998, xxi, 350. ISBN 0387984364. info
Teaching methods
Lectures: theoretical explanation. Exercises: solving problems with the aim to understand basic concepts and theorems, homework.
Assessment methods
Examination consists of two parts: a written test and an oral examination. To pass the written part it is necessary to get at least 50% of points. The students successful in the written part have to show in the following oral part that they are able to define the used notions and to work with them, to formulate the explained statements and to prove the easier of them.
Language of instruction
Czech
Further Comments
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Spring 2003, Spring 2005, Spring 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022.
  • Enrolment Statistics (recent)
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