MATHS 320 : Algebraic Structures

Science

Course Prescription

This is a framework for a unified treatment of many different mathematical structures. It concentrates on the fundamental notions of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Applications abound: symmetries, geometry, coding theory, cryptography and many more. This course is recommended for those planning graduate study in pure mathematics.

Course Overview

This course is intended for students who have enjoyed the examples of algebraic structures presented in MATHS 254/255, 253 and/or MATHS 328, and who are interested in a unified treatment of many different mathematical structures. The main focus of the course is on the fundamental algebraic structures of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Emphasis is placed on the contexts in which basic structures occur, methods by which they can be modelled and analysed, and their diverse applications. This course is recommended for those planning graduate study in pure mathematics. A student completing this course will have a basic knowledge of the fundamental of algebraic systems, and be able to construct simple proofs in an algebraic setting and appreciate the strength the strength of abstract algebraic methods.

Course Requirements

Prerequisite: MATHS 250, and MATHS 254 or 255

Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity Capability 6: Social and Environmental Responsibilities

Learning Outcomes

By the end of this course, students will be able to:
1. Display basic knowledge of fundamental algebraic systems: groups, rings and fields and their properties (Capability 1 and 2)
2. Display ability to analyse and answer questions, through examples and methods (Capability 1, 2 and 3)
3. Demonstrate understanding of connections with other disciplines and real world applications by the use of relevant examples (Capability 1, 2, 3 and 4)
4. Demonstrate understanding of the nature of proof and precision in mathematics by presenting informal arguments in both verbal and written form (Capability 1, 2, 4 and 5)
5. Engage in critical interactions (Capability 2, 3, 4 and 6)

Assessments

Assessment Type Percentage Classification
Tutorials 10% Individual Coursework
Assignments 20% Individual Coursework
Mid-semester Test 25% Individual Test
Final Exam 45% Individual Examination
1 2 3 4 5
Tutorials
Assignments
Mid-semester Test
Final Exam

You must achieve at least 35% in the final exam in order to pass this course.

Key Topics

Week 1: Symmetric operations and groups
Week 2: Subgroups (including cyclic subgroups)
Week 3: Cyclic groups; Permutation groups
Week 4: Group isomorphisms; Direct products; Cosets
Week 5: Lagrange’s Theorem; Normal subgroups and factor groups
Week 6: Group homomorphisms; Finite abelian groups
Week 7: Rings and integral domains
Week 8: Ideals, factor rings and ring homomorphisms
Week 9: Polynomial rings and factorisation
Week 10: Vector spaces (including extension fields)
Week 11: Field extensions (including algebraic extensions)
Week 12: Finite fields.

Learning Resources

The recommended textbook is "Contemporary Abstract Algebra", by J. A. Gallian (published by D.C. Heath & Company), 8th edition or later, but also detailed course notes will be provided, and these will include definitions, examples, properties and applications of the key concepts developed in the course.

Special Requirements

This course has no special requirements.

This course is a standard 15-point course, and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in, making a total of 150 hours.

For this course, you can expect 36 hours of lectures and 11 hours of tutorials, and you should spend approximately 36 hours of reading and thinking about the content, and 67 hours of work on assignments and/or test preparation.

Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 09/07/2020 05:04 p.m.