Mathematical Studies
MA335: Algebraic Structures (5 ECTS)
An introduction to the theory and application of modern abstract algebra.
Taught in Semester(s) 2. Examined in Semester(s) 2.
Workload: 126 hours (24 Lecture hours, 12 Tutorial hours, 90 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Compute products, inverses and signs of permutations. Use permutations to calculate symmetry groups, find inverses of integers modulo n, multiply quaternions, factor polynomials, encode and decode using finite fields.
- Determine fundamental regions for lattices, calculate the volume of a fundamental region using determinants, express numbers as sums of squares.
- Present to a class an outline of a real world application of the course material.
- Write proofs of standard theorems in the area of modern abstract algebra.
Indicative Content
Syllabus Outline:
- Groups Rings and Fields
Permutation groups, Symmetry groups, Units of the integers modulo $n$, group axioms, Quaternions, Polynomial rings, Gaussian integers, Finite fields, ring and field axioms, Applications. (14 lectures) - Lattices and Lattice Points in Euclidean Space Fundamental regions and their volumes, Minkowski's Theorem, primes that are sums of two squares, non-negative integers as sums of four squares. (7 lectures)
- Fermat's Last Theorem Pythagorean triples and their classification, Fermat (The Story) and his Last Theorem ($n=4$), outline of modern developments via Geometry. (3 lectures)
Module Resources
- "Elementary Number Theory" by G.A. Jones and J.M. Jones, Springer
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