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
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.
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 2017, Spring 2019, Autumn 2020.
  • Enrolment Statistics (Spring 2015, recent)
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