Groups Rings and Fields
Course category:2nd year
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.
Extension fields. Splitting fields of polynomials; constructing finite fields; the group of units of a finite field is cyclic.
Group decompositions. Internal and external direct and semidirect products.
Group actions. Actions and the Sylow theorems, illustrations.
J B Fraleigh, A First Course Abstract Algebra, Addison-Wesley.
Three hour closed examination in mid-Summer term.
A second course on abstact algebra, assuming a basic knowledge of group theory. 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 much of the material taught in the modules Core Algebra and all of that in Introduction to Group Theory, or are willing to learn the material if necessary.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD