Module Content

The module covers
  • Introduction to Group Theory,
  • Introduction to Rings and Fields,
  • Introduction to Number Theory,
  • and explains how these topics can be applied to
  • The symmetries of a regular polygon,
  • Cryptography.

Module Coordinates

  • Lecturer: Angela Carnevale
  • Lectures: Monday 12-1 in AC204, Wednesday 12-1 in AC213
  • Tutorials: Wednesday 3-4 in AM121
  • Recomended text: "A First Course in Abstract Algebra", John B. Fraleigh (available at 512.02 FRA)
  • Problem sheet: available here.
  • Module Website: Information and module documents will be posted to this site, which is linked from the Blackboard MA335 Algebraic Structures pages. Blackboard will also be used for announcements and for posting grades.

Module Assessment

  • End of semester examination: 50%.
  • Continuous assessment: 30%.
  • Communications skils: 20%.

Supplementary Material and News

A model paper is avaialble here.

Lecture Notes

Lecture Notes
(click on number)

Lecture Summaries
Introduction to the course. Symmetries as a motivation to define and study groups. Rotations and symmetries of regular polygons and their compositions.
Definition of a group. Basic properties and examples.
Permutation groups: definition, two-line and cycle notation.
Permutations as products of transpositions. Even and odd permutations. Cayley tables. Subgroups: definition, basic facts and examples.
The even permutations form a subgroup of the symmetric group. Integral powers of group elements and cyclic subgroups. Examples.
More on cyclic subgroups and their orders. Order and divisibility: subgroups of S_4. Have a look at the Cayley table of S_4 and of its subgroups here.
More on subgroups of S_4 and A_4. Lagrange's theorem. Structural theorem for cyclic groups. The integers modulo n as additive groups.
Briefing on Communication Skills. Group homomorphisms and isomorphisms. The quaternion group. Cayley's theorem.
Introduction to rings: definition, basic properties and examples. Zero divisors and integral domains.
First test.
Rings, subrings. The ring of integers modulo n. Units of a ring.
Fields: definition and examples. A field is an integral domain. A finite integral domain is a field. Zero divisors of integers modulo n and finite fields. Characteristic of a ring. Fermat's little theorem.
Weekly recap. Euclidean algorithm to find gcd(a,b) and gcd(a,b) as linear combination of a and b. Euler's totient (phi) function and its properties. Units modulo n and Euler's theorem.
More on Euler's theorem and the extended Euclidean algorithm. The RSA algorithm.
Second test. Here you can find guidelines for your essay. We will discuss this in class too.
Polynomial rings, the division algorithm. Zeros of polynomials and linear factors.
Greatest common divisor and Euclidean algorithm in F[x]. Irreducible polynomials.
Unique Factorisation Theorem for polynomials. Irreducibility over the reals and over the complex numbers. Irreducibility of polynomials of small degree over a field. Irreducibility over the rationals and over the integers: Gauss's Lemma and its corollaries.
Eisenstein's criterion for irreducibility. Examples. Application to the p-th cyclotomic polynomials. Introduction to field extensions.
The Gaussian integers: definition, properties, division algorithm. Brief introduction to Euclidean domains and UFDs.
Third test.
Communications skills: presentations.
Revision and exam preparation.