Instructor: Dr. Áron BERECZKY
Text: Peter J. Cameron: Introdction to Algebra (Oxford Univerity Press) 1998. chapters 1, 2, 3, sections 7.1.1, 7.1.2, 7.2.1.
The course provides an introduction to ring theory and group theory.
The methods correspond to an introductory level.
Introduction: relations, functions, operations, polynomials, matrices.
Elementary ring theory: rings, subrings, ideals, factor rings.
Factorization in rings: 0-divisors, units, irreducibles, factorization, Euclidean domains, PID, UFD and the connection between them.
Fields: maximal ideals in rings, quotient fields, field of fractions, existence of simple extensions
Elementary group theory: properties of groups, subgroups, cosets, Lagrange's theorem, cyclic groups, order of an element.
Homomorphisms: Normal subgroups, factor groups, isomorphism theorems, conjugacy.
permutations: Cayley's theorem, symmetric and alternating
groups, group actions and permutation groups, orbit, stabilizer,
groups of small order, symmetry groups, Sylow's theorems.