Syllabus
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.
Annotation
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).