PURE MTH 7072 - Fields and Modules
North Terrace Campus - Semester 2 - 2020
General Course Information
Course Code PURE MTH 7072 Course Fields and Modules Coordinating Unit School of Mathematical Sciences Term Semester 2 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Assumed Knowledge PURE MTH 2106 and PURE MTH 3007 Course Description This subject presents the foundational material for the last of the basic algebraic structure 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.
Course Coordinator: Dr Daniel Stevenson
The full timetable of all activities for this course can be accessed from Course Planner.
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 structures. 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.
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s) Deep discipline knowledge
- informed and infused by cutting edge research, scaffolded throughout their program of studies
- acquired from personal interaction with research active educators, from year 1
- accredited or validated against national or international standards (for relevant programs)
1, 2, 3, 4, 5 Critical thinking and problem solving
- steeped in research methods and rigor
- based on empirical evidence and the scientific approach to knowledge development
- demonstrated through appropriate and relevant assessment
1, 2, 3, 4, 5, 6 Teamwork and communication skills
- developed from, with, and via the SGDE
- honed through assessment and practice throughout the program of studies
- encouraged and valued in all aspects of learning
Recommended ResourcesStudents may wish to consult any of the following books, available in the Library.
M. Artin, “Algebra”.
J. A. Beachy, “Introductory lectures on rings and modules”.
J. B. Fraleigh, “A first course in abstract algebra”.
B. Hartley, T. O. Hawkes, “Rings, modules and linear algebra”.
I. N. Herstein, “Topics in algebra”.
S. Lang, “Algebra”.
S. Lang, “Undergraduate algebra”.
R. Y. Sharp, “Steps in commutative algebra”.
Online LearningAssignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
S. Lang, “Undergraduate algebra”, is available as an e-book via the Library catalogue.
Learning & Teaching Activities
Learning & Teaching ModesThe lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload hours Lectures 30 90 Tutorials 6 18 Quizzes ongoing 18 Assignments 5 30 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 University'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 task Task type Due Weighting Learning outcomes Examination Summative Examination period 70% All Homework assignments Formative and summative Weeks 3, 5, 7, 9, 10, 12 10% All Mid-semester Test Formative and summative Lecture 15 10% All Quizzes Formative and summative ongoing 7% All Tutorial participation Formative and summative Weeks 2, 4, 6, 8, 10,12 3% All
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment task Set Due Weighting Assignment 1 Week 1 Week 2 4% Tutorial exercises 1 Week 2 Week 3 see below Assignment 2 Week 3 Week 4 4% Tutorial exercises 2 Week 4 Week 5 Assignment 3 Week 5 Week 6 4% Tutorial exercises 3 Week 6 Week 7 Assignment 4 Week 7 Week 8 4% Tutorial exercises 4 Week 8 Week 9 Assignment 5 Week 9 Week 10 4% Tutorial exercises 5 Week 10 Week 11 Assignment 6 Week 11 Week 12 4%
SubmissionHomework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.
Grades for your performance in this course will be awarded in accordance with the following scheme:
M10 (Coursework Mark Scheme) Grade Mark Description FNS Fail No Submission F 1-49 Fail P 50-64 Pass C 65-74 Credit D 75-84 Distinction HD 85-100 High Distinction CN Continuing NFE No Formal Examination RP Result Pending
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 Access Adelaide.
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