35003 Modern Algebra
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particular session, location and mode of offering is the authoritative source
of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.
Subject handbook information prior to 2024 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 or Algebra 68105
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
32019 Directed Study 1,
32009 Directed Study 2,
32021 Directed Study 3,
or
31013 Directed Study 4 if in Computer Science/FEIT Faculty. Please discuss with your Program Coordinator in Maths or FEIT.
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.
Workshops are a mix of content delivery and working in small groups collaboratively on problems. Frequent short learning progress checks throughout the semester will assess progress, understanding and skills. The final assignment will cover all topics 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: Class presentations
| 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 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: | Presentation |
| Groupwork: | Group, group and individually assessed |
| Weight: | 25% |
| Criteria: | Correctness of mathematics, clarity of exposition. |
Assessment task 2: Short learning progress checks
| 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 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: | 40% |
| Criteria: | Correct mathematics, clarity of exposition, appropriate choice of mathematical techniques/proof methods. |
Assessment task 3: Final summative assessment task
| 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 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: | Exercises |
| Groupwork: | Individual |
| Weight: | 35% |
| 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".