#
PřF:M7230 Galois Theory - Course Information

## M7230 Galois Theory

**Faculty of Science**

Spring 2015

**Extent and Intensity**- 3/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). 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 **Timetable**- Thu 13:00–15:50 M2,01021
**Prerequisites**(in Czech)- Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
**Course Enrolment Limitations**- The course is offered to students of any study field.
**Course objectives**- Lecture on Galois theory including some of its applications in algebra and geometry. 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**- Field extension: simple algebraic extension, the degree of extension, algebraic and transcendental extension.
- Constructibility by straightedge and compas: imposibility to construct solution of the following geometric problems posed by the Greeks: doubling the cube, trisecting an angle, squaring the circle (without a proof that "pi" is transcendental).
- Normal and separable extension, linear independence of the embeddings of a field, normal closure, Galois correspondence.
- Solvable and simple groups.
- Solvability of algebraic equations in radicals: radical extensions.
- Unified view on solutions of quadratic, cubic and biquadratic equations, construction of an equation of degree five insolvable in radicals over the field of rational numbers.
- Galois group of cyclotomic fields, constructibility of regular polygons by straightedge and compas.

**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 - PROCHÁZKA, Ladislav.
*Algebra*. Vyd. 1. Praha: Academia, 1990. 560 s. ISBN 8020003010. info

- DUMMIT, David Steven and Richard M. FOOTE.
**Teaching methods**- Lectures: theoretical explanation with applications in 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 (probably available only in Czech)**- Study Materials

The course is taught once in two years.

- Enrolment Statistics (Spring 2015, recent)
- Permalink: https://is.muni.cz/course/sci/spring2015/M7230