35003 Modern Algebra
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Subject handbook information prior to 2025 is available in the Archives.
Credit points: 8 cp
Result type: Grade and marks
There are course requisites for this subject. See access conditions.
Recommended studies:
Discrete Mathematics 37181, Linear Algebra 37233, Algebra 68105
Description
This subject develops students ability to think abstractly, crucial for advanced mathematics and computer science. Students explore groups, rings, and fields through practical applications, including cryptography. Topics cover permutations, group theory, finite fields, and polynomial factoring. For 6 credit points, Science Faculty students can enrol in '35391 Seminar (Mathematics)', while Computer Science/FEIT students may enrol in '32019/32009/32021/31013 Directed Study'. Consult the relevant Program Coordinator for permission.
Subject learning objectives (SLOs)
Upon successful completion of this subject students should be able to:
| 1.0. | Demonstrate practical and theoretical skills in abstract algebra. |
|---|---|
| 2.0. | Identify and evaluate approaches to solve problems, including in collaboration with others. |
| 3.0. | Participate in group-based discussions, working effectively and responsibly in a group |
| 4.0. | Construct mathematical proofs, make conjectures, construct counterexamples, develop mathematical/logical creativity, reflect on whether a proof is correct, if it could be done more efficiently, if it could be made more general. |
| 5.0. | Present written and oral solutions to problems using appropriate presentation and information. |
Course intended learning outcomes (CILOs)
This subject also contributes specifically to the development of following course intended learning outcomes:
- Demonstrate appraisal and interpretation of mathematical and statistical principles and concepts, with a focus on applying methodologies to solve complex problems. (1.1)
- Tackle the challenge of complex real-world problems in the areas of mathematical and statistical modelling by evaluating and analysing information and solutions. (2.1)
- Engage in work practices that demonstrate an understanding of confidentiality requirements, ethical conducts, data management, and organisation and collaborative skills in the context of applying mathematical and statistical modelling. (3.1)
- Find and reflect the value, integrity, and relevance of multiple sources of information to derive creative solutions, innovation and application of technologies in evaluating solutions to contemporary mathematics problems. (4.1)
- Identify, present complex ideas and justifications using appropriate communication approaches from a variety of methods (oral, written, visual) to communicate with mathematicians, data analysts, scientists, industry, and the general public. (5.1)
Contribution to the development of graduate attributes
1.0 Disciplinary knowledge
Students will learn and be assessed on practical and theoretical skills in modern algebra.
2.0 Research, inquiry and critical thinking
Students will learn and be assessed on skills in idenitifying and evaluating alternative approaches to solving problems.
3.0 Professional, ethical, and social responsibility
Students will learn and be assessed on how to work effectively and responsibly in a group during the workshops.
4.0 Reflection, innovation, creativity
Students will learn and be assessed on proof writing, making conjectures, constructing counterexamples in algebra.
5.0 Communication
Students will learn and be assessed on how to present written and oral solutions to problems using appropriate professional language.
Teaching and learning strategies
Students should attend and actively participate in the two workshops each week, and spend several hours each week working on homework problems outside of class time.
Workshops are a mix of content delivery and working in small groups collaboratively on problems.
The subject will cover Chapters 2-4 of Lauritzen, textbook available without cost from UTS Library (online version). Additional materials and assessments will be available on Canvas.
Content (topics)
In this subject we will study the foundations of modern algebra: groups, rings, fields, polynomial rings. To this end, we will cover:
- axiomatic definition of a group
- proving theorems about groups, how to classify finite groups
- Sylow theorems
- rings and fields
- principal ideal domains, Euclidean domains
- polynomial rings
- classification of finite fields
Assessment
Assessment task 1: Assessment item 1
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1.0 disciplinary knowledge 2.0 research, inquiry and critical thinking 4.0 reflection, innovaton, creativity 5.0 communication. |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1.0, 2.0, 3.0, 4.0 and 5.0 This assessment task contributes to the development of course intended learning outcome(s): 2.1, 4.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 45% |
| Length: | 1 hour 10 minutes for 1A; 1 hour 10 minutes for 1B |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Assessment task 2: Assessment item 2
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1.0 disciplinary knowledge 2.0 research, inquiry and critical thinking 4.0 reflection, innovaton, creativity 5.0 communication. |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1.0 and 5.0 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 3.1, 4.1 and 5.1 |
| Type: | Quiz/test |
| Groupwork: | Individual |
| Weight: | 35% |
| Length: | 1 hour 10 minutes |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Assessment task 3: Assessment item 3
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1.0 disciplinary knowledge 3.0 professional, ethical, and social responsibility 4.0 reflection, innovation, creativity 5.0 communication |
|---|---|
| Objective(s): | This assessment task addresses subject learning objective(s): 1.0, 2.0 and 5.0 This assessment task contributes to the development of course intended learning outcome(s): 1.1, 2.1 and 5.1 |
| Type: | Presentation |
| Groupwork: | Group, individually assessed |
| Weight: | 20% |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Required texts
Lauritzen, N. (2003). Concrete abstract algebra : from numbers to Gro?bner bases. Cambridge University Press.
Available online without cost from the UTS Library website: https://search.lib.uts.edu.au/permalink/61UTS_INST/19joism/alma991004425689705671
Recommended texts
Fraleigh, John B. (2014). A first course in abstract algebra. Seventh edition, Pearson new international edition.
Available (online readable) without cost from the UTS Library website: https://search.lib.uts.edu.au/permalink/61UTS_INST/19joism/alma991007395272305671