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Kurzovní přednáška z algebry pro navazující magisterské učitelské studium (polynomy a jejich kořeny, Lagrangeova
postupná symetrizace; přechod v algebře od hledání kořenů polynomů ke zkoumání struktur). Propojení algebraických
témat se školskou matematikou (diskriminant, Vietovy věty, zavedení komplexních čísel, různé způsoby řešení
kvadratické rovnice).
Last update: Robová Jarmila, doc. RNDr., CSc. (04.06.2020)
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Successful completion of a written test (120 minutes). It is necessary to demonstrate an understanding of all the topics discussed in the lecture. Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (29.10.2019)
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Basic literature:
Dlab V., Bečvář J.: Od aritmetiky k abstraktní algebře. Serifa, Praha, 2016.
Additional literature:
Bewersdorff J.: Galois Theory for Beginners; A Historical Perspective. Student Mathematical Library (Book 35), AMS, 2006. 180 stran.
Tignol J.-P.: Galois' Theory of Algebraic Equations. World Scientific Publishing, Singapore, 2001.
Blažek J. a kol.: Algebra a teoretická aritmetika I, II. SPN, Praha, 1983, 1984.
Stanovský D.: Základy algebry. Matfyzpress, Praha, 2010. Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (04.06.2020)
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Elementary introduction to Galois theory
Solution of quadratic and cubic equations by different methods, comparison of methods applicable in school mathematics. Viete's formulas.
Elementary introduction to Galois theory, Lagrange's symmetrization, application of Viete's formulas, symmetric polynomials, cyclic groups, factorization of permutation groups.
Symmetric polynomials and discriminant
Simple and elementary symmetric polynomials. Relation to Viete's formulas. Discriminant - general definition and its calculation, connection with school mathematics.
Polynomials and fields
Comparison of different definitions of a polynomial and their application in school mathematics. Elimination of root multiplicity, derivation of a polynomial. Boundaries for polynomial roots. Horner's scheme. Lagrange's interpolation.
Relationship between Q[x] and Z[x], examples, Eisenstein's criterion.
Primitive field, finite field structure. Algebraic field closure.
Introduction of complex numbers in school mathematics, Kronecker's theorem. Field extension, splitting fields, examples.
Solvability of algebraic equations in radicals, Galois correspondences.
Groups and their classification
Simple, cyclic, abelian groups - examples and contexts. A_5 is simple, the consequences. Cauchy's theorem. Sylow's theorems and their applications. Last update: Halas Zdeněk, Mgr., DiS., Ph.D. (04.06.2020)
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