MATH216 - Abstract Algebra and Equations

Year

Credit points

Campus offering

Prerequisites

MATH107 Introduction to Logic and Algebra

Incompatible

MATH303 Abstract Algebra 

Teaching organisation

Unit rationale, description and aim

This unit extends the study of algebra started in MATH107 and introduces abstract algebra - one of the main threads of mathematics. Groups and fields are introduced as tools that may be used to solve polynomial equations. The historical background to abstract algebra is considered in this context. To support this aim the unit will cover such topics as: subgroups, normal subgroups, symmetries, field extensions and irreducibility in the context of polynomials. Galois theory will be mentioned but not assessed.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Demonstrate knowledge of the historical background of abstract algebra, in particular the solution of polynomial equations (GA5) 

LO2 - Demonstrate an understanding of the concepts of groups, subgroups (GA5) 

LO3 - Identify and use normal subgroups (GA5) 

LO4 - Demonstrate an understanding of the concepts of fields and field extensions (GA5) 

LO5 - Describe the relationship between solutions of polynomial equations and field extensions (GA5) 

LO6 - Solve polynomial equations, in particular cubic equations, using techniques involving symmetries (GA5, GA7, GA8, GA10) 

LO7 - Demonstrate knowledge of the impossibility of solving quintic equations via radicals (GA5, GA8). 

Graduate attributes

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA7 - work both autonomously and collaboratively 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics may include: 

1. Solutions of polynomial equations – history, techniques and symmetric polynomials 
2. Definition of groups, motivated from symmetries of plane geometrical figures. Permutation notation and introduction to subgroups 
3. Cycle notation and examples of groups including:
4. Normal subgroups and homomorphisms 
5. Lagrange’s theorem 
6. Polynomials and irreducibility, Eisenstein’s criterion 
7. Polynomial equations and an introduction to fields 
8. Field extensions and irreducible polynomials 
9. Discussion of cubic, quartic and quintic equations 
10. The connection between groups and field extensions, finding approximate solutions to polynomials including complex solutions 

Learning and teaching strategy and rationale

This unit includes 4 contact hours per week over 12 weeks, comprising 2 hours of lectures and 2 of tutorials. 

Assessment strategy and rationale

A range of assessment procedures will be used to meet the unit learning outcomes and develop graduate attributes consistent with University assessment requirements. Such procedures may include, but are not limited to: essays, reports, examinations, student presentations or case studies. 

The assessment will include an extended group-work task which will specifically address the learning outcome (6) and graduate attributes GA7, GA8 and GA10. 

Overview of assessments

Representative texts and references

Cooke, R. (2008). Classical algebra: Its nature, origins and uses. Hoboken, NJ: John Wiley & Sons. 

Dummit, D., & Foote, R. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: John Wiley & Sons. 

Jordan, D. (1994). Groups. Oxford: Butterworth-Heinemann. 

Maxfield, J., & Maxfield, M. (2010). Abstract algebra and solution by radicals. Mineola, NY: Dover Publications. 

Pinter, C. (2010). A book of abstract algebra (2nd ed.). Mineola, NY: Dover Publications. 

Rotman, J. (1994). An introduction to the theory of groups. London: Springer-Verlag. 

Stewart, I. (2004). Galois theory (3rd ed.). London: Chapman and Hall.