Syllabus for Algebraic Structures

Learning outcomes

On completion of the course, the student should be able to:

• give an account of important concepts and definitions for groups, rings and fields;
• exemplify and interpret important concepts in specific cases;
• formulate important results and theorems covered by the course;
• describe the main features of the proofs of important theorems;
• express problems from relevant areas of applications in a mathematical form suitable for further analysis;
• use the theory, methods and techniques of the course to solve problems about groups, rings and fields;
• present mathematical arguments to others.

Content

The group concept. Isomorphisms and homomorphisms. Subgroups and residue classes. The order of a group element, cyclic groups. Normal subgroups, quotient groups. Group action on a set, orbit, stabiliser, conjugation. Solvable groups and Sylow theorems. Abelian groups. Classification of finitely generated Abelian groups.

The ring concept. Isomorphisms and homomorphisms. Subring, ideal and quotient ring. Invertible elements, maximal ideals. Constructions of non-commutative rings: rings of matrices over arbitrary rings, rings of operators, sub rings of such rings, the ring of quaternions. Irreducible elements, prime elements, prime ideals and principal ideals in commutative rings. Euclidean rings. Unique factorisation for Euclidean rings. Factorial rings: greatest common divisor and least common multiple, Gauss theorem. Irreducibility criteria for polynomials over factorial rings: Gauss lemma, Eisenstein's criterion.

The field concept. The group of automorphisms. Finite fields. Field extensions. Algebraic and transcendental extensions. Separable and normal extensions. The Galois group. The fundamental theory of Galois theory. Solvability of algebraic equations. Formulas for third and fourth degree equations. Geometric construction problems.

Instruction

Lectures and problem solving sessions.

Assessment

Written and oral examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.