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 also required a deep understanding of how 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 subgroups, normal subgroups, homomorphisms, symmetries, field extensions and irreducibility in the context of polynomials. The basic ideas of Galois theory, one of the highest and most beautiful achievement of modern algebra will be considered at an introductory level. 

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.

2026 10

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Prerequisites

MATH107 Introduction to Logic and Algebra AND (MATH205 Geometry OR MATH220 Linear Algebra OR MATH222 Number Theory and Cryptography )

Incompatible

MATH216 Abstract Algebra and Equations , MATH303 - Abstract Algebra

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.

Demonstrate knowledge of the historical background...

Learning Outcome 01

Demonstrate knowledge of the historical background of abstract algebra, in particular the solution of polynomial equations
Relevant Graduate Capabilities: GC1, GC3, GC7, GC8

Use the concepts of groups and subgroups to solve ...

Learning Outcome 02

Use the concepts of groups and subgroups to solve simple problems and prove simple results. Identify and use normal subgroups, particularly with reference to group homomorphisms.
Relevant Graduate Capabilities: GC1, GC7, GC8

Use the concepts of fields and field extensions to...

Learning Outcome 03

Use the concepts of fields and field extensions to solve simple problems. Describe the relationship between solutions of polynomial equations and field extensions. Solve polynomial equations, in particular cubic equations, using techniques involving symmetries
Relevant Graduate Capabilities: GC1, GC7, GC8

Content

Topics will include:

  1. Solutions of polynomial equations – history, techniques and symmetric polynomials
  2. Definition of groups
  3. Cycle notation and examples of groups
  4. Lagrange’s theorem
  5. Normal subgroups and group 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

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.

At specified times, through the semester, students will sit two supervised components to ensure that students have fully integrated the learning and can independently bring a variety of strategies to solve a variety of problems. The take home assignment will allow students to develop other skills including traditional and technological problem-solving methods.

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, the lecturer may request that assignments and tests are handwritten

Overview of assessments

Assessment Task 1: Take Home Assignment(s) Conti...

Assessment Task 1: Take Home Assignment(s)

Continuous assessment – A single task which is submitted in 1, 2 or 3 parts across the semester as stipulated by the lecturer.

Weighting

30%

Learning Outcomes LO1, LO2, LO3
Graduate Capabilities GC1, GC3, GC7, GC8

Assessment Task 2: Test Content and application...

Assessment Task 2: Test

Content and application test 1. This test will be conducted under supervision and on campus.

Weighting

35%

Learning Outcomes LO1, LO2
Graduate Capabilities GC1, GC3, GC7, GC8

Assessment Task 3: Test Content and application ...

Assessment Task 3: Test

Content and application test 2. This test will be conducted under supervision and on campus.

Weighting

35%

Learning Outcomes LO1, LO2, LO3
Graduate Capabilities GC1, GC3, GC7, GC8

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 through tutorials that ideally occur face to face. Face-to-face learning supports immediate feedback, peer interaction, and collaborative problem-solving (essential elements in developing mathematical thinking) while also enabling tutors to identify and address misconceptions early.

Continuous formative learning opportunities allow for the early detection of problems a student might be having and ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be expected to submit written responses to questions involving core mathematical concepts, methods, and problem-solving strategies. In all cases this should be supported using available online technology.

Representative texts and references

Representative texts and references

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

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

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

Fraleigh, J.B. & Brand, N. (2020) A First Course in Abstract Algebra (8th ed.), Pearson

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

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

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

Rotman, J.J. (2014). An introduction to the theory of groups, Springer Verlag

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

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

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