PURE MTH 3023 - Fields and Modules III

North Terrace Campus - Semester 2 - 2019

This subject presents the foundational material for the last of the basic algebraic structures 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.

  • General Course Information
    Course Details
    Course Code PURE MTH 3023
    Course Fields and Modules III
    Coordinating Unit School of 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
    Assumed Knowledge PURE MTH 2106, PURE MTH 3007
    Course Description This subject presents the foundational material for the last of the basic algebraic structures 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 Staff

    No information currently available.

    Course Timetable

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

  • Learning Outcomes
    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 Galois theory to the theory of polynomial equations
    6 Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
    7 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
    6
  • Learning Resources
    Required Resources
    The course is based on the book "Undergraduate Algebra" by Serge Lang. This is available from the library as an e-book.
    Recommended Resources
    None
    Online Learning
    Assignments, 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 Modes
    The 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 and on-going online quizzes help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
    Workload

    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.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

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

    Assessment Summary
    Assessment taskTask typeDueWeightingLearning 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 Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    Assessment taskSetDue
    Assignment 1 Week 2 Week 3
    Tutorial exercises 1 Week 1 Week 2
    Assignment 2 Week 4 Week 5
    Tutorial exercises 2 Week 3 Week 4
    Assignment 3 Week 6 Week 7
    Tutorial exercises 3 Week 5 Week 6
    Assignment 4 Week 8 Week 9
    Tutorial exercises 4 Week 7 Week 8
    Assignment 5 Week 10 Week 11
    Tutorial exercises 5 Week 11 Week 12
    Assignment 6 Week 12 Week 13
    Each assignment will count for roughly 1.6% of the final grade.  The tutorial participation grade will be determined by, well, participation in tutorials.  This may include presenting problems to the class and/or active participation in discussion.  
    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 University'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:

    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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    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 University 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.

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  • Policies & Guidelines
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