MATH316 - Abstract Algebra and Equations

Year

Credit points

Campus offering

Prerequisites

MATH107 Introduction to Logic and Algebra AND (MATH205 Geometry or MATH220 Linear Algebra or MATH221 Applications of Mathematics or MATH222 Number Theory and Cryptography or MATH223 History and Philosophy of Mathematics )

Incompatible

MATH216 Abstract Algebra and Equations , MATH303

Unit rationale, description and aim

Despite its name, abstract algebra was developed in order to solve specific problems, in particular to solve polynomial equations. The solution to those problems required a deep understanding of how, for example, number systems worked. Any study of mathematics, beyond a purely introductory level, requires that deep understanding of the structure of number systems and other algebraic constructs.

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. The unit will cover such topics as: subgroups, normal subgroups, homomorphisms, symmetries, field extensions and irreducibility in the context of polynomials. Galois theory will be mentioned but not assessed.

The aim of this unit is to give students an appreciation of why a study of algebraic systems, including number systems, is important for solving certain problems. To support this aim, students will be introduced to two of main abstract algebraic structures: groups and fields.

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 - Use the concepts of groups, subgroups to solve simple problems and prove simple results (GA5)

LO3 - Identify and use normal subgroups, particularly with reference to group homomorphisms (GA5)

LO4 - Use the concepts of fields and field extensions to solve simple problems (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 - Provide an outline explanation 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 will 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
4. Lagrange’s theorem
5. Normal subgroups and homomorphisms
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

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology.

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial.

150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study.

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems in abstract algebra. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students.

The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure.

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit.

The continuous assessment task will include specific questions that require students to solve cubic or quartic polynomials.

Overview of assessments

Representative texts and references

Alcock, L. (2021). How to Think About Abstract Algebra Oxford: Oxford University Press.

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.J. (2014). An introduction to the theory of groups. New York: Springer Verlag

Stewart, I. (2015). Galois theory (4th ed.). Oakville: Apple Academic Press Inc.

Whitelaw, T.A. (2018) An Introduction to Abstract Algebra 3rd Edition. London: Taylor & Francis