Algebraic Structures

10 credits

Syllabus, Bachelor's level, 1MA007

A revised version of the syllabus is available.

Code: 1MA007
Education cycle: First cycle
Main field(s) of study and in-depth level: Mathematics G2F
Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
Finalised by: The Faculty Board of Science and Technology, 15 March 2007
Responsible department: Department of Mathematics

Entry requirements

Algebra I, Linear Algebra II. Algebra II recommended.

Learning outcomes

In order to pass the course (grade 3) 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. Irreducible elements, prime elements, prime ideals and principal ideals in commutative rings. Euclidean rings. Unique factorisation for Euclidean rings. Irreducibility criteria for integral polynomials.

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, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.