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MT4517 RINGS AND FIELDS
AimsThe overall aim of the course is to provide an introduction to commutative ring theory. Historically, rings have appeared as generalisations of number systems (integers, in particular) with the intention of gaining deeper insight into number systems themselves. This will be reflected in this course, where students will study familiar concepts, such as factorisation, primeness, divisibility etc., in a new, more general, setting of commutative rings.
In addition, the course may include topics from: rings of quotients, finite fields and extensions of fields.
ObjectivesBy the end of the course students are expected to be able to
- write elements of a factorisation domain as products of irreducibles;
- understand the connection between primes and irreducibles in an arbitrary integral domain;
- investigate whether an integral domain is a unique factorisation domain. When it is not, to be able to find essentially different factorisations of a given element and to prove the factorisations essentially different;
- for an integral domain which is not a unique factorisation domain, to be able to find essentially different factorisations of a given element and to prove the factorisations essentially different;
- find greatest common divisors and least common multiples and to decide when they are unique (up to associates);
- work with rings of polynomials over finite fields;
- work with the ring Z[n];
- understand the construction of the ring of quotients of an integral domain and its connection with the construction of the rational numbers.
Syllabus- Fermat's Last Theorem and historical introduction. Rings, subrings, subrings of the complex numbers. Fields, division rings and integral domains. Ideals, prime ideals, maximal ideals.
- Factorisation in integral domains. Prime elements and irreducible elements. The division algorithm.
- Uniqueness of quotients. Greatest common divisors and least common multiples. Principal ideal domains, factorisation domains, unique factorisation domains. Norms and Euclidean rings.
- Gaussian integers. Ring of integers of a quadratic number field. Some discussion of prime ideals and factorisation of ideals into products of prime ideals. Finite fields.
TextbooksR B J T Allenby, Rings, Fields and Groups, 1991 [Now out of print, but on reserve in library].
Groups, Rings and Fields, Algebra Through Practice Book 3: T S Blyth and E F Robertson; Cambridge University Press.
Assessment2 Hour Examination = 100%
PrerequisitesMT3600 or (MT2002 and MT3501)
AvailabilityAcademic year 2012/13 in semester 1 at 11
LecturerDr M Neunhöffer, Dr J D Mitchell
Click here for access to past examination papers for this module.
Click here to see the University Course Catalogue entry.
Revised: PMH (September 2012)