Groups Rings and Fields
Course category:2nd year
Group actions. Actions and the Sylow theorems, illustrations.
Rings. Definition of rings and examples (including matrix rings and rings of functions), consequences of axioms, subrings.
Fundamental theorem of ring homomorphisms. Homomorphisms, ideals, quotient rings, kernels and homomorphism theorems.
Special rings. Integral domains, principal ideal domains and fields.
Ideals. Prime and maximal ideals: for commutative R, R/I is an integral domain if and only if I is prime; R/I is a field if and only if I is maximal.
Division in commutative rings with identity; unique factorization domains; Euclidean rings. Examples including polynomial rings. Euclidean rings are PIDs, PIDs are UFDs.
Division. Further examples.
Extension fields. Splitting fields of polynomials; constructing finite fields; the group of units of a finite field is cyclic.
J B Fraleigh, A First Course Abstract Algebra, Addison-Wesley.
Three hour closed examination.
A second course on abstact algebra, assuming a basic knowledge of group thoery. Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD