35003 Modern Algebra
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Subject handbook information prior to 2023 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 and Linear Algebra 37233.
Description
This subject enables students to think abstractly and work confidently with concepts underpinning much of modern mathematics and computer science. Abstract concepts such as groups, rings and fields are introduced via non-trivial applications and examples which give real-world motivation. Topics include factoring algorithms for integers, quadratic residues, permutations, normal subgroups and simple groups, finite fields, principal ideal domains and unique factorization domains, factoring algorithms for polynomials; applications to modern (post-quantum) cryptography.
Note: undergraduate students who want to take Modern Algebra as a 6 credit point subject can enrol in 35391 Seminar (Mathematics) if in Science Faculty and 31013 Directed Study if in Computer Science/FEIT Faculty.
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:
- Synthesise: Integrate extensive knowledge of sub-disciplines of the Mathematical Sciences providing a pathway for further learning and research. (1.3)
- Analyse: Make arguments based on proof and independently conduct simulations based on identified approaches (e.g. analytic vs numerical/experimental, different statistical tests, different heuristic algorithms) and various sources of data and knowledge. (2.2)
- Apply: Work effectively and responsibly in an individual or team context with advanced professional and interpersonal skills. (3.1)
- Analyse: Advanced information retrieval and consolidation skills applied to the critical evaluation and analysis of the mathematical/statistical aspects of information to think creatively and try different approaches to solving problems. (4.2)
- Analyse: Conduct advanced independent research to clarify a problem or to obtain the information required to develop elegant mathematical solutions. (5.2)
- Synthesise: Integrate written and verbal instructions or problem statements to describe a significant complex piece of work and its importance, and place the work in the context of other scholarly research. (5.3)
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.
Workshops will include content delivery and working in small groups collaboratively on problems.
The two midterm assessments will assess progress and skills gained during the semester, and the final assessment will bring together all topics covered and again assess progress and skills gained during the semester.
The subject will cover Chapters 1-4 of Lauritzen. 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:
- background on number theory, quadratic residues
- 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
Assessment
Assessment task 1: Presenting homework problems
| Intent: | This assessment task contributes to the development of the following graduate attributes: 2.0 research, inquiry and critical thinking 4.0 reflection, innovaton, creativity 5.0 communication. |
|---|---|
| Objective(s): | This assessment task contributes to the development of course intended learning outcome(s): 1.3, 2.2, 3.1 and 4.2 |
| Type: | Presentation |
| Groupwork: | Group, group and individually assessed |
| Weight: | 20% |
| Criteria: | Correctness of mathematics, clarity of exposition. |
Assessment task 2: Midterm tests
| 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 contributes to the development of course intended learning outcome(s): 1.3, 2.2, 3.1, 4.2 and 5.2 |
| Type: | Mid-session examination |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Assessment task 3: Assignment
| Intent: | This assessment task contributes to the development of the following graduate attributes: 1.0 disciplinary knowledge 2.0 research, inquiry and critical thinking 5.0 communication |
|---|---|
| Objective(s): | This assessment task contributes to the development of course intended learning outcome(s): 1.3, 2.2, 4.2, 5.2 and 5.3 |
| Type: | Exercises |
| Groupwork: | Individual |
| Weight: | 40% |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Minimum requirements
Students are strongly recommended to attend and actively participate in each workshop.
Required texts
Lauritzen, N. (2003). Concrete Abstract Algebra: From Numbers to Gröbner Bases. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511804229
Check the UTS Library website for access to hardcopies or PDF via "Cambridge Core".