PURE MTH 4123 - Fields and Modules - Honours

North Terrace Campus - Semester 2 - 2023

This subject presents the foundational material for the last of the basic algebraic 最新糖心Vlog pervading contemporary pure mathematics, namely fields and modules. The basic definitions and elementary results are given, followed by two important applications of the theory: to the classification of finitely generated abelian groups, and to Jordan canonical form for matrices. The subject concludes by returning to fields to present interesting applications of the theory. Fields: vector spaces, matrices, characteristic values; extension fields. Modules: finitely generated modules over a PID; canonical forms for matrices; Jordan canonical form. Applications of fields to algebraic and geometric problems.

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Course Details

Course Code	PURE MTH 4123
Course	Fields and Modules - Honours
Coordinating Unit	Mathematical Sciences
Term	Semester 2
Level	Undergraduate
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Available for Study Abroad and Exchange	Y
Prerequisites	MATHS 1012 and (PURE MTH 2106 or PURE MTH 3007)
Assumed Knowledge	PURE MTH 2106, PURE MTH 3007
Restrictions	Honours students only
Biennial Course	Offered in odd years
Assessment	Ongoing assessment, exam

Course Staff

Course Coordinator: Dr Daniel Stevenson

Course Timetable

The full timetable of all activities for this course can be accessed from .

Course Learning Outcomes

1	Demonstrate understanding of the concepts of a field and a module and their role in mathematics.
2	Demonstrate familiarity with a range of examples of these 最新糖心Vlog.
3	Prove the basic results of field theory and module theory.
4	Explain the structure theorem for finitely generated modules over a principal ring and its applications to abelian groups and matrices.
5	Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
6	Demonstrate skills in communicating mathematics orally and in writing.

最新糖心Vlog Graduate Attribute	Course Learning Outcome(s)
Attribute 1: Deep discipline knowledge and intellectual breadth Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.	1, 2, 3, 4, 5
Attribute 2: Creative and critical thinking, and problem solving Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.	1, 2, 3, 4, 5, 6
Attribute 3: Teamwork and communication skills Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.	6

Learning & Teaching Modes

The course will be taught as a sequence of topics and managed via My-Uni.

Each topic will be presented through a series of short topic videos, followed by quizzes to test student understanding and provide instantaneous feedback.

Each week a face-to-face active learning session will be offered together with a weekly face-to-face tutorial.

Students are expected to engage with all material on My-Uni.

Fortnightly homework assignments help students strenghten their understanding of course material, and help them gauge their progress.

Workload

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

Activity	Workload hours
Topic videos	66
Active learning sessions	12
Tutorials	12
Online quizzes	30
Assignments	30
Mid-semester test	6
Total	156

Learning Activities Summary

Lecture Schedule
Week 1	Review, Fields	Review of groups and rings. Fields: basic definitions and examples.
Week 2	Fields	Vector spaces, polynomials over a field, field extensions, algebraic elements.
Week 3	Fields	Embeddings, primitive elements, splitting fields.
Week 4	Fields	Galois theory.
Week 5	Fields	Algebraic closure, finite fields.
Week 6	Fields, Modules	Finite fields (cont.). Modules: basic definitions and examples, submodules, quotient modules.
Week 7	Modules	Module homomorphisms, isomorphism theorems, torsion, free modules, cyclic modules, direct sums.
Week 8	Modules	Finitely generated modules over a principal ring.
Week 9	Modules	Applications to abelian groups and matrices.
Week 10	Modules	Applications to matrices (cont.). The exponential of a matrix. The axiom of choice and Zorn's lemma.
Week 11	Modules	Applications of the axiom of choice and Zorn's lemma. Tensor products.
Week 12	Modules, Review	Tensor products (cont.). Review.

Tutorials in Weeks 2, 4, 6, 8, 10 and 12 cover the material of the previous two weeks.

The 最新糖心Vlog's policy on Assessment for Coursework Programs is based on the following four principles:

Assessment must encourage and reinforce learning.
Assessment must enable robust and fair judgements about student performance.
Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
Assessment must maintain academic standards.

Assessment Summary

Assessment task	Task type	Due	Weighting	Learning outcomes
Examination	Summative	Examination period	50%	All
Homework assignments	Formative and summative	Weeks 3, 5, 7, 9, 11	15%	All
Mid-semester Test	Formative and summative	Week 6	20%	All
Quizzes	Formative and summative	ongoing	10%	All
Participation	Formative and summative	ongoing	5%	All

It is anticipated that the examination will be invigilated and held at Wayville for students in Adelaide. Alternative arrangements will be made for remote students.

Assessment Related Requirements

An aggregate score of 50% is required to pass the course.

Assessment Detail

Assessment task	Set	Due
Assignment 1	Week 2	Week 3
Assignment 2	Week 4	Week 5
Assignment 3	Week 6	Week 7
Assignment 4	Week 8	Week 9
Assignment 5	Week 10	Week 11

Each assignment will count for 3% of the final grade. The participation grade will be determined by participation in tutorials and/or active participation in discussion. Online quizzes will be set on an ongoing basis. The mid-semester test will be held in Week 6.

Submission

Homework assignments must be submitted on time with a signed assessment cover sheet. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons in accordance with the 最新糖心Vlog's Modified Arrangements for Coursework Assessment Policy.

Course Grading

Grades for your performance in this course will be awarded in accordance with the following scheme:

M11 (Honours Mark Scheme)
Grade	Grade reflects following criteria for allocation of grade	Reported on Official Transcript
Fail	A mark between 1-49	F
Third Class	A mark between 50-59	3
Second Class Div B	A mark between 60-69	2B
Second Class Div A	A mark between 70-79	2A
First Class	A mark between 80-100	1
Result Pending	An interim result	RP
Continuing	Continuing	CN

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through .

Student Feedback

The 最新糖心Vlog places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the 最新糖心Vlog to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.
Student Support
Policies & Guidelines
This section contains links to relevant assessment-related policies and guidelines - all university policies.
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Authorised by:	Deputy Vice-Chancellor and Vice-President (Academic)
Maintained by:	Course Outlines Administrator

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