(Created 2011-09-01.)
 ALGEBRA FMA190
Credits: 6. Grading scale: TH. Cycle: A (Second Cycle). Main field: Technology. Language of instruction: The course might be given in English. Optional for: C4, C4ks, D4, D4ks, E4, E4ks, F3, Pi2, Pi2pv, Pi2ssr. Course coordinator: Director of Studies Anders Holst, Anders.Holst@math.lth.se, Mathematics. Recommended prerequisits: In principle Calculus and Linear algebra are sufficient, but the greater mathematical maturity provided by further mathematical courses is helpful . Assessment: Written and/or oral test, to be decided by the examiner.

Aim
The aim of the course is to give an introduction to the basic concepts of abstract algebra, with particular regard to subjects of importance in applications in, e.g., computer science, information theory, physics and chemistry. The course also aims to give a deeper understanding of the basic concepts in other areas of mathematics. Furthermore, the course should develop the student's problemsolving ability and her/his ability to understand mathematical text.

Knowledge and understanding
For a passing grade the student must

be able to describe basic properties of integers and polynomials, and be able to compute with congruences modulo these objects.

be able to describe basic properties of the important concepts in abstract algebra; ring, ideal, quotient ring, group and field.

in writing and orally be able to explain the contents of some central definitions and proofs.

be able to give examples of and illustrate some important applications of the course content.

Skills and abilities
For a passing grade the student must

be able to independently construct proofs of simple statements within the framework of the course.

be able to show a good ability to independently, in writing and orally, explain mathematical reasoning in a well structured way, with clear logic.

Contents
Rings: Polynomial rings. Ideals and quotient rings. Ring homomorphisms and isomorphisms.

Groups: Lagrange's theorem. Permutation groups. Normal subgroups and quotient groups. Group homomorphisms and isomorphisms.

Fields: Characteristic. Finite fields. Field extensions.

Literature
Hungerford: Abstract Algebra, an introduction, 2nd ed. Brooks/Cole 1996. ISBN 0-03-010559-5