MAS2006: Groups, Rings and Fields

Lectures: 3pm Mondays, 12pm Tuesdays, 11am Fridays, in C95 for 11 weeks, starting Monday 7 October 2002.

Lecturer: Robin Chapman, room 811, rjc@maths.ex.ac.uk, web page:

http://www.maths.ex.ac.uk/∼rjc/rjc.html ,

course web page:

http://www.maths.ex.ac.uk/∼rjc/courses/grf02/grf02.html .

Brief syllabus: Groups: Symmetry and the concept of a group. Subgroups, cosets, Lagrange’s theorem. Cyclic groups. Permutation groups, matrix groups, symmetry groups of geometrical objects. Isomorphisms and homomorphisms. Kernel and image. Normal subgroups and quotient groups. Arithmetical applications. Rings & Fields: Review of properties of standard number systems. Examples of rings: integers modulo n, polynomial rings, rings of quadratic integers, rings of functions. Units and zero-divisors, integral domains. Isomorphisms and homomorphisms. Kernel and image. Factorization, primes, irreducibles. Euclidean domains. Non-commutative examples: matrix rings, quaternions.

Recommended Book: John B. Fraleigh, A First Course in Abstract Algebra.

Assessed Course Work (25%): Example sheets due in Mondays weeks 3, 5, 7, 9, and 11 of semester. Examination (75%): 2 hours, week 13 of semester.