Information
Timetable:
| Monday | 10.00-11.00 | (lecture) | Richard Berry-213 |
| Wednesday | 9.00-10.00 | (prac class) | Alice Hoy-101 |
| Wednesday | 10.00-11.00 | (lecture) | Richard Berry-213 |
| Thursday | 10.00-11.00 | (lecture) | Richard Berry-213 |
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: applications to number theory, the classical impossibility theorems, and existence of quintic polynomialsin Q[x]whose roots can not be expressed by radicals over Q.
Books:
The main references are:
Algebra; M. Artin, Prentice-Hall (1991)
(For rings, fields and Galois theory. A very nice book.)Rings, modules and linear algebra; B. Hartley and T. O. Hawkes, Chapman and Hall (1970)
(For modules. Artin's book covers modules, but we shall follow the more contructive approach to the Structure Theorem given in Hartley and Hawkes)
Other references include:
Abstract algebra, an introduction; T. W. Hungerford, Saunders College Publishing (1990)
(Good for brushing up on background material.)A first course in abstract algbra; J. B. Fraleigh (1973)
(I like this book a lot. Covers many more topics than are in this subject.)Algebra; S. Lang, Adison-Wessley (1965)
(A classic.)
Assessment:
Written assignments (20 percent) and 3 hour examination (80 percent).
Prerequisite:
620-222 (prior to 2009) with a grade of H3 or better.
Handbook entry:
is here.
Generic skills:
A statement can be found here.