Introduction to Abstract Algebra — ALG1

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.

Prerequisite: ---

Course description:
The course provides an introduction to ring theory and group theory.
The methods correspond to an introductory level.

Topics:
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.

Group actions, permutations: Cayley's theorem, symmetric and alternating groups, group actions and permutation groups, orbit, stabilizer, groups of small order, symmetry groups, Sylow's theorems.