Goal: Acquirement of basic algebraic and number theoretic notions and theorems, their application in exercises.
Course description: Operations, algebraic structures, basics of group theory, permutation groups, Cayley theorem, Lagrange theorem, normal subgroups, factor groups, homomorphisms, Isomorphism theorems, Sylow theorems, simple groups, soluble groups, nilpotent groups, Abelian groups, composition series, direct products, fundamental theorem of finite Abelian groups; free groups, basics of ring theory, commutative rings, ideals, factor rings, principal ideal domains, Noetherian rings, integral domains, fields, construction of fields, finite fields, field extensions, modules, algebras, basics of number theory, fundamental theorem of arithmetic, Euclidean algorithm, Euclidean rings, principal ideal domains, unique factorization domains. Number theory in polynomial rings, polynomials with integral coefficients: Gaussian lemma, criterion of Schönemann and Eisenstein. Congruence, linear congruences, Euler’s totient functions, quadratic congruences. Basic concepts of Lie algebras.
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