# 35003 Modern Algebra

**Warning:** The information on this page is indicative. The subject outline for a
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 2023 is available in the Archives.

*UTS: Science: Mathematical and Physical Sciences*

*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. |
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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)
- Apply: Succinct and accurate presentation of complex information, reasoning, and conclusions in a variety of modes to diverse expert and non-expert audiences. (5.1)
- 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. |
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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): 2.2, 4.2 and 5.1 |

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 |
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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): 1.3, 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 |
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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): 1.3, 2.2, 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".