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.

M7230 Galois Theory

Faculty of Science
Autumn 2022
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer), Mgr. Pavel Francírek, Ph.D. (deputy)
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
Wed 10:00–11:50 M6,01011
  • Timetable of Seminar Groups:
M7230/01: Thu 16:00–17:50 M3,01023, R. Kučera
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
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.
  • 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 2024.

M7230 Galois Theory

Faculty of Science
Autumn 2020
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer), Mgr. Pavel Francírek, Ph.D. (deputy)
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 16:00–17:50 M3,01023
  • Timetable of Seminar Groups:
M7230/01: Fri 8:00–9:50 M3,01023, R. Kučera
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
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.
  • 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 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2019
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)
Mgr. Pavel Francírek, Ph.D. (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
Mon 18. 2. to Fri 17. 5. Wed 8:00–10:50 M3,01023
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
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
  • 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
  • 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
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
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, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2017
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
Mon 20. 2. to Mon 22. 5. Wed 10:00–12:50 M6,01011
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
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
  • 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
  • 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
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
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 2019, Autumn 2020, Autumn 2022, Autumn 2024.

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, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2013
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 8:00–10:50 M5,01013
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 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2011
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
Timetable
Tue 8:00–10:50 M6,01011
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
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:
understand main results on Galois theory;
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: written test and oral examination.
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 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2009
Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Timetable
Wed 8:00–10:50 M1,01017
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
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:
understand main results on Galois theory;
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
Assessment methods
Standard lecture. Examination consists of two parts: written and oral.
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 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2007
Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Timetable
Wed 8:00–10:50 N41
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • DUMMIT, David Steven and Richard M. FOOTE. Abstract algebra. 3rd ed. Hoboken, N.J.: John Wiley & Sons, 2004, xii, 932. ISBN 0471433349. info
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
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 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2005
Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Timetable
Mon 15:00–17:50 UP1
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
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 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2003
Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Spring 2005, Spring 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2025

The course is not taught in Spring 2025

Extent and Intensity
2/2. 6 credit(s). Type of Completion: zk (examination).
In-person direct teaching
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
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
  • 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
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
Course is no more offered.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2024

The course is not taught in Spring 2024

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
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
  • 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
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
Course is no more offered.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Autumn 2023

The course is not taught in Autumn 2023

Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer), Mgr. Pavel Francírek, Ph.D. (deputy)
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
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
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.
  • 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
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2023

The course is not taught in Spring 2023

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
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
  • 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
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
Course is no more offered.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2022

The course is not taught in 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
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
  • 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
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.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
autumn 2021

The course is not taught in autumn 2021

Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Radan Kučera, DSc. (lecturer), Mgr. Pavel Francírek, Ph.D. (deputy)
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
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.
  • 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
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2021

The course is not taught in Spring 2021

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
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
  • 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
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.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2020

The course is not taught in Spring 2020

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
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
  • 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
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.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
spring 2018

The course is not taught in spring 2018

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
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
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
  • 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
  • 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
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.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2016

The course is not taught in Spring 2016

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
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
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)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2014

The course is not taught in Spring 2014

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
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)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2012

The course is not taught in Spring 2012

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
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:
understand main results on Galois theory;
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: written test and oral examination.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2010

The course is not taught in Spring 2010

Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
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:
understand main results on Galois theory;
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: written test and oral examination.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2008

The course is not taught in Spring 2008

Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Guaranteed by
prof. RNDr. Radan Kučera, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • DUMMIT, David Steven and Richard M. FOOTE. Abstract algebra. 3rd ed. Hoboken, N.J.: John Wiley & Sons, 2004, xii, 932. ISBN 0471433349. info
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2006

The course is not taught in Spring 2006

Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2004

The course is not taught in Spring 2004

Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2011 - only for the accreditation
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
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
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:
understand main results on Galois theory;
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: written test and oral examination.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Spring 2003, Spring 2005, Spring 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Autumn 2020, Autumn 2022, Autumn 2024.

M7230 Galois Theory

Faculty of Science
spring 2012 - acreditation

The course is not taught in spring 2012 - acreditation

The information about the term spring 2012 - acreditation is not made public

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
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:
understand main results on Galois theory;
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: written test and oral examination.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.

M7230 Galois Theory

Faculty of Science
Spring 2008 - for the purpose of the accreditation

The course is not taught in Spring 2008 - for the purpose of the accreditation

Extent and Intensity
3/0. 3 credit(s) (fasci plus compl plus > 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
Contact Person: prof. RNDr. Radan Kučera, DSc.
Prerequisites (in Czech)
Algebra II (tj. odborná) nebo Algebra 2 (tj. učitelská)
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
  • Mathematics (programme PřF, M-MA, specialization Discrete Mathematics)
Course objectives (in Czech)
Cílem přednášky je výklad Galoisovy teorie a jejích některých aplikací v algebře i geometrii.
Syllabus (in Czech)
  • Rozšíření teles: jednoduché algebraické rozšíření, stupeň rozšíření, algebraické a transcendentní rozšíření. Konstrukce pravítkem a kružítkem: nemožnost konstrukce úloh zdvojení krychle, trisekce úhlu a kvadratury kruhu (bez důkazu, že $\pi$ je transcendentní). Normální a separabilní rozšíření, lineární nezávislost vnoření těles, normální uzávěr, Galoisova korespondence. Řešitelné a jednoduché grupy. Řešitelnost algebraických rovnic v radikálech: radikálová rozšíření. Jednotný pohled na řešení rovnic kvadratických, kubických a rovnic čtvrtého stupně, konstrukce rovnice pátého stupně neřešitelné v radikálech nad racionálními čísly. Galoisova grupa kruhových teles, konstrukce pravidelných mnohoúhelníků pravítkem a kružítkem.
Literature
  • DUMMIT, David Steven and Richard M. FOOTE. Abstract algebra. 3rd ed. Hoboken, N.J.: John Wiley & Sons, 2004, xii, 932. ISBN 0471433349. info
  • PROCHÁZKA, Ladislav. Algebra. Vyd. 1. Praha: Academia, 1990, 560 s. ISBN 8020003010. info
  • STEWART, Ian. Galois theory. 2nd ed. London: Chapman & Hall, 1989, xxx, 202 s. ISBN 0-412-34550-1. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
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, Autumn 2024.
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