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PřF:M7230 Galois Theory - Course Information

## M7230 Galois Theory

**Faculty of Science**

Spring 2022

**Extent and Intensity**- 2/2. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- prof. RNDr. Radan Kučera, DSc. (lecturer)
**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 **Prerequisites**(in Czech)- 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**- Algebra and Discrete Mathematics (programme PřF, N-MA)
- Geometry (programme PřF, N-MA)

**Course objectives**- Lecture on Galois theory including some of its applications in algebra and geometry.
**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.
- Infinite Galois theory.

**Literature****Teaching methods**- Lectures: theoretical explanation with applications to concrete examples.
**Assessment methods**- Examination consists of two parts: a written test and an oral examination. To pass the written part, which consists of 7 exercises, 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**- The course is taught once in two years.

The course is taught: every week.

- Enrolment Statistics (Spring 2022, recent)
- Permalink: https://is.muni.cz/course/sci/spring2022/M7230