3036.18 - Algebra
Mathematics 1 (3103)
The aim of the course is to introduce the student to the basic concepts and methods of abstract algebra, with emphasis on elementary number theory, groups, rings, fields, and polynomials.
Arithmetic, prime numbers, prime factorization, elementary number theory, roots of unity, mathematical induction, groups, subgroups, homomorphisms and isomorphisms of groups, cyclic groups, permutation groups, Lagrange’s theorem, Fermat’s little theorem, finite abelian groups, the fundamental theorem for finite abelian groups, normal subgroups, factor groups, rings and fields, ideals of rings, quotient rings, homomorphisms and isomorphisms of rings, field of fractions, polynomials, polynomial rings, roots of polynomials.
Learning and teaching approaches
Lectures and exercise classes. Written assignments with feedback.
By the end of the course the student is expected to be able to: • Recognize group and ring structures, and analyze simple aspects of these. • Reproduce definitions and results covered in the course in a mathematically rigorous way. • Apply algebraic techniques, concepts, and results to analyze concrete examples of algebraic structures. • Write simple mathematical proofs. • Perform modular arithmetic computations. • Factorize polynomials over different rings. • Determine kernels and images of homomorphisms.
The grade will be determined by weekly written assignments and a 4 hours written examination which will take place at the end of the course. Reexamination is a 4 hours written examination, where the student chooses either to reuse the written assignments or make new submissions. All written material and calculators will be allowed at the examination, no other electronic devices are allowed.
Anders Thorup: ”Algebra. Matematik 2AL” (1995-version). ISBN 87-91180-08-2 (2006, 4. oplag)
Toke Meier Carlsen