Algebraic Structures

Entry requirements

60 credits in mathematics. Algebra I. Linear Algebra II. Algebra II.

Learning outcomes

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

• give an account of important concepts and definitions for groupsand 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 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 actions on sets, orbit, stabiliser, conjugation. Burnside's lemma. Solvable groups. Sylow theorems. Abelian groups. Classification of finitely generated Abelian groups. Free groups and presentations of groups.

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. Geometric construction problems.

Instruction

Lectures and problem solving sessions.

Assessment

Written exam (9 hp). Oral examination (1 hp).

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.