Information
Prerequisite:
620-222 Linear and Abstract Algebra (grade of H3 or better)
Timetable:
Monday 12.00-1.00 (lecture), Richard Berry-213
Tuesday 4.15-5.15 (practice class), Richard Berry-G03
Wednesday 10.00-11.00 (lecture), Engineering-E-317 [Theatre E2]
Thursday 11.00-12.00 (lecture), Engineering-E-317 [Theatre E2]
There is no prac class in the first week.
Topics:
This subject provides further experience with abstract algebraic concepts and methods. General structural results are proved and algorithms developed to determine the invariants they describe. The material covered is widely used in algebraic topology and in number theory.
Rings: abstract rings and isomorphisms; examples including matrix rings and polynomial rings; homomorphisms, ideals and quotient rings; integral domains; units, irreducibles and primes; prime and maximal ideals; field of quotients; Euclidean domains and principal ideal domains.
Modules: submodules, homomorphisms of modules, quotient modules; free modules and bases; structure of a finitely generated module over a principal ideal domain; applications to abelian groups and to Jordan normal form of matrices.
Fields: field extensions and their construction; the degree of a field extension; finite fields; Galois extensions, splitting fields and the Galois correspondence.
Additional topics may include tensor and exterior algebras, applications to number theory, the classical impossibility theorems, and structure theory for simple rings.
References used include:
A first course in abstract algbra; J. B. Fraleigh (1973)
Abstract algebra, an introduction; T. W. Hungerford, Saunders College Publishing (1990)
Rings, modules and linear algebra; B. Hartley and T. O. Hawkes, Chapman and Hall (1970)
Algebra; M. Artin, Prentice-Hall (1991)
Basic Algebra I; N. Jacobson, W. H. Freeman (1974)
Algebra; S. Lang, Adison-Wessley (1965)
Assessment:
Written assignments (20 percent) and 3 hour examination (80 percent).
Generic skills:
A statement can be found here.
Handbook entry:
is here.