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.
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.