Algebra - NMUM501
Title: Algebra
Guaranteed by: Department of Mathematics Education (32-KDM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 4
Hours per week, examination: winter s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Additional information: http://www.karlin.mff.cuni.cz/~halas/Algebra.htm
Guarantor: Mgr. Zdeněk Halas, DiS., Ph.D.
Incompatibility : NMTM501
Interchangeability : NMTM501
Is incompatible with: NMTM501, NUMP020
Is interchangeable with: NMTM501, NUMP020
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Annotation -
This course builds on the pre-graduate course Arithmetic and algebra I and II. Elements of Galois theory, polynomials, symmetric polynomials, groups, fields, ruler and compass constructibility.
Last update: Staněk Jakub, RNDr., Ph.D. (14.06.2019)
Course completion requirements -

Written test:

reach the point score at least 80% of the highest possible score and at the same time it must be clear that the student generally understands each of the topics.

Attendance at seminars is not obligatory.

The necessary condition for passing the exam is to pass the written test.

Exam:

The subject matter is in the range of lectures and exercises (if concrete examples illustrate an idea significant in theory),

possibly also in the range of texts assigned for independent repetition or study.

Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (29.10.2019)
Literature -

Basic literature:

Dlab V., Bečvář J.: Od aritmetiky k abstraktní algebře. SERIFA, Praha, 2016. 480 stran.



Supplementary literature:

Bewersdorff J.: Galois Theory for Beginners; A Historical Perspective. Student Mathematical Library (Book 35), American Mathematical Society, 2006. 180 stran.

Pesic P.: Abel's Proof: An Essay on the Sources and Meaning of Mathematical Unsolvability. MIT Press, 2004. 222 stran.

Chajda I.: Vybrané kapitoly z algebry. UP, Olomouc, 2014. 72 stran.

Stanovský D.: Základy algebry. Matfyzpress, Praha, 2010.

Laubenbacher R., Pengelley D.: Mathematical Expeditions; Chronicles by the Explorers. Springer, New York, 1999. (kap. 5.3, 5.4 a 5.5)

Livio M.: Neřešitelná rovnice; Matematika a jazyk symetrií. Argo/Dokořán, 2008.


Blažek J. a kol.: Algebra a teoretická aritmetika I. SPN, Praha, 1983.

Blažek J. a kol.: Algebra a teoretická aritmetika II. SPN, Praha, 1985.


Katriňák T. a kol.: Algebra a teoretická aritmetika I. Alfa, Bratislava, 1985.

Šalát T. a kol.: Algebra a teoretická aritmetika II. Alfa, Bratislava, 1986.

Hungerford T. W.: Algebra (Graduate Texts in Mathematics). Springer, 2003.

Knapp A. W.: Basic Algebra. Along with a companion volume 'Advanced Algebra'. Birkhäuser, Basel, 2006.

Vinberg E. B.: A Course in Algebra. Graduate Studies in Mathematics, AMS, 2003.

Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (14.06.2019)
Requirements to the exam -

The examination consists only of the oral part.

The exam can be taken after obtaining the credit.

The requirements of the exam correspond to the syllabus of the subject to the extent that was presented at the lecture, including everything that was ordered for individual study.

Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (29.10.2019)
Syllabus -
Polynomials:
  • Solution of quadratic, cubic and quartic equation.
  • Multiple roots, derivative of a polynomial, distribution of roots of a polynomial with real coefficients.
  • Symetric polynomials.

Groups:

  • Elements of group theory, Abel and cyclic groups, Theorem of Cauchy and Sylow.

Fields:

  • Field extensions, solvability.
  • Ruler and compass constructibility.
Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (14.06.2019)