Course syllabus

# Algebraic StructuresAlgebraiska strukturer

## FMAN10, 7.5 credits, A (Second Cycle)

Valid for: 2024/25
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2024-04-15
Effective: 2024-05-08

## General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: C4-sec, D4, F4, Pi4-pv
Language of instruction: The course will be given in English on demand

## Aim

The aim of the course is to give an introduction to the fundamental concepts and structures of abstract algebra, with particular regard to subjects of importance in applications in, e.g., computer science, information theory, physics and chemistry. The course also aims to give a deeper understanding of the basic concepts in other areas of mathematics. Furthermore, the course should develop the students' ability to solve problems and to understand mathematical text.

## Learning outcomes

Knowledge and understanding
For a passing grade the 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 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 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.

Judgement and approach
For a passing grade the student must

## Contents

Number theory: The fundamental theorem of arithmetic, modular arithmetic.

Rings: Definition and basic properties. Polynomial rings. Ideals and quotient rings. Ring homomorphisms and isomorphisms.

Groups: Definition and basic properties. Normal subgroups and quotient groups. Group homomorphisms and isomorphisms. Lagrange's theorem. Permutation groups. Symmetric and alternatic groups. Finitely generated Abelian groups.

Fields: Characteristic. Finite fields. Field extensions.

## Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment: Written examination followed, for those who pass, by an oral examination.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0113. Name: Algebraic Structures.
Credits: 7.5. Grading scale: TH - (U, 3, 4, 5).

## Admission

Assumed prior knowledge: In terms of content, the courses in Calculus and Linear algebra are sufficient. However, without the greater mathematical maturity provided by one or more further courses in mathematics it is difficult to pass the course.
The number of participants is limited to: No
Kursen överlappar följande kurser: FMA190 MATM11 MATM31

## Reading list

• Hungerford, T W: Abstract Algebra - An Introduction. Brooks/Cole, 2012, ISBN: 9781111573331. 3rd edition. Since this edition is very expensive the literature may be changed. Otherwise the 2nd edition may be used.

## Contact

Course coordinator: Anders Holst, Anders.Holst@math.lth.se
Teacher: Anna Torstensson, anna.torstensson@math.lth.se
Course administrator: Studerandeexpeditionen, expedition@math.lth.se
Course homepage: https://canvas.education.lu.se/courses/20590