Course Syllabus for

# Algebraic Structures Algebraiska strukturer

## FMAN10F, 7.5 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-01-27

## General Information

Division: Mathematics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course codes: FMAN10, MATM11
Teaching language: English

## Aim

The course is established in order to give the doctoral students knowledge about basic concepts of abstract algebra, which is useful for further studies in mathematics as well as within digital technology and coding theory.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to describe basic properties of integers and polynomials, and be able to compute with congruences modulo these objects.
• be able to describe basic properties of the important concepts in abstract algebra; ring, ideal, quotient ring, group and field.
• be able to explain, in writing and orally, the contents of some central definitions and proofs.
• be able to give examples of and illustrate some important applications of the course contents.
• have acquired basic knowledge for further studies in algebra or subjects based on algebraic methods.

Competences and Skills

For a passing grade the doctoral student must

• For a passing grade the student must
• be able to independently construct proofs of simple statements within the framework of the course.
• be able to show a good ability to independently, in writing and orally, explain mathematical reasoning in a well structured way, with clear logic.

## Course Contents

Rings: Polynomial rings. Ideals and quotient rings. Ring homomorphisms and isomorphisms. Groups: Lagrange's theorem. Permutation groups. Normal subgroups and quotient groups. Group homomorphisms and isomorphisms. Fields: Characteristic. Finite fields. Field extensions.

## Course Literature

Hungerford, T.: Abstract Algebra: An Introduction. Cengage Learning, 2012. ISBN 9781111573331.

## Instruction Details

Types of instruction: Lectures, exercises

## Examination Details

Examination formats: Written exam, oral exam