Rings & Fields

Description

Many examples of rings will be familiar before entering this course. Examples include the integers modulo n, the complex numbers and n-by-n matrices with real entries. The course develops from the fundamental definition of ring to study particular classes of rings and how they relate to each other. We also encounter generalisations of familiar concepts, such as what is means for a polynomial to be prime. Basic concepts and examples. Ideals, factor rings, isomorphism theorems. Rings of polynomials. Field of fractions of a domain. Unique Factorization Domains, Principal Ideal Domains, Euclidean Domains. Passage from R to R[X]. Gauss's Theorem. Eisenstein's criterion. Fields : characteristic, prime subfield. Finite fields, construction. Algebraic and transcendental elements, algebraic closure. Syllabus Rings Zero divisors and integral domains. Homomorphisms. Ideals and quotient rings. Field of fractions of an ID. ID, UFD, PID and ED. Polynomial rings over commutative rings. Field extensions Splitting fields. Finite fields.