- Class Number 3560
- Term Code 3030
- Class Info
- Unit Value 6 units
- Mode of Delivery In Person
- Dr James Borger
- Dr James Borger
- Class Dates
- Class Start Date 24/02/2020
- Class End Date 05/06/2020
- Census Date 08/05/2020
- Last Date to Enrol 02/03/2020
Just as there is a formula for solving a quadratic equation, there are similar formulae for solving the general cubic and quartic. Galois theory provides a solution to the corresponding problem for quintics --- there is no such formula in this case! Galois theory also enables us to prove (despite regular claims to the contrary) that there is no ruler and compass construction for trisecting an angle. More broadly, the purpose of Galois theory is to study polynomials at a deep level by using symmetries between the roots. This is a pervasive theme in modern mathematics, and Galois theory is traditionally where one first encounters it.
Topics to be covered include:
Galois Theory - fields, field extensions, normal extensions, separable extensions. Revision of group theory, abelian and soluble groups.The main theorem of Galois theory.Solubility of equations by radicals. Finite fields. Cyclotomic fields.
Note: This is an HPC. It emphasises mathematical rigour and proof and continues the development of modern analysis from an abstract viewpoint.
Upon successful completion, students will have the knowledge and skills to:
On satisfying the requirements of this course, students will have the knowledge and skills to:
1. Explain the fundamental concepts of field extensions and Galois theory and their role in modern mathematics and applied contexts
2. Demonstrate accurate and efficient use of field extensions and Galois theory
3. Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts from field extensions and Galois theory
4. Apply problem-solving using field extensions and Galois theory applied to diverse situations in physics, engineering and other mathematical contexts.
Staff FeedbackStudents will be given feedback in the following forms in this course:
- Written comments
- Verbal comments
- Feedback to the whole class, to groups, to individuals, focus groups
Student FeedbackANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.
|Week/Session||Summary of Activities||Assessment|
|2||Maximal and prime ideals, polynomials in one variable, field extensions|
|3||Adjoining elements to fields, Gauss's lemma, Eisenstein's criterion|
|4||Algebraic and transcendental elements, minimal polynomials, tower degree formula, algebraic numbers, algebraic extensions, algebraically closed fields, cyclotomic fields|
|5||Ruler and compass constructions and their impossibility theorems, splitting fields|
|6||Homomorphisms of extensions, isomorphy of splitting fields, separable and inseparable polynomials|
|7||Frobenius morphisms, perfect fields, finite fields, automorphism group of extensions|
|8||Automorphism groups as permutation groups, separable and normal extensions, Galois extensions, fixed fields|
|9||Fundamental theorem of Galois theory and its proof|
|10||Galois theory of finite and cyclotomic fields, cyclotomic polynomials|
|11||Constructibility of regular polygons, discriminant subfield, radical extensions, solvable groups|
|12||Solvability by radicals, additional topics|
Workshops by arrangement
|Assessment task||Value||Due Date||Return of assessment||Learning Outcomes|
|Mid-semester exam||35 %||30/03/2020||24/04/2020||1,2,3,4|
|Final examination||35 %||04/06/2020||02/07/2020||1,2,3,4|
* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details
PoliciesANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:
Assessment RequirementsThe ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the ANU Online website Students may choose not to submit assessment items through Turnitin. In this instance you will be required to submit, alongside the assessment item itself, hard copies of all references included in the assessment item.
Moderation of AssessmentMarks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.
Please note, that where a date range is used in the Assessment Summary in relation to exams, the due date and return date for mid-semester exams indicate the approximate timeframe in which the exam will be held; the due and return date for end of semester exams indicate the approximate timeframe in which the exam will be held and the date official end of Semester results are released on ISIS. Students should consult the course wattle site and the ANU final examination timetable to confirm the date, time and venue of the exam.
Assessment Task 1
Learning Outcomes: 1,2,3,4
There are 5 assignments due over the semester worth 6% each (total 30%). It is intended that the marked assignments will be returned within 1 week after submission. Further details can be found on the Course Wattle site.
Assessment Task 2
Learning Outcomes: 1,2,3,4
Please check the course Wattle site and the ANU Examination Timetable to confirm the date, time and location of the mid semester exam.
Assessment Task 3
Learning Outcomes: 1,2,3,4
The date range in the Assessment Summary indicates the start of the end of semester exam period and the date official end of semester results are released on ISIS. Please check the ANU final Examination Timetable http://www.anu.edu.au/students/program-administration/assessments-exams/examination-timetable to confirm the date, time and location exam.
Academic IntegrityAcademic integrity is a core part of our culture as a community of scholars. At its heart, academic integrity is about behaving ethically. This means that all members of the community commit to honest and responsible scholarly practice and to upholding these values with respect and fairness. The Australian National University commits to embedding the values of academic integrity in our teaching and learning. We ensure that all members of our community understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with. The University has policies and procedures in place to promote academic integrity and manage academic misconduct. Visit the following Academic honesty & plagiarism website for more information about academic integrity and what the ANU considers academic misconduct. The ANU offers a number of services to assist students with their assignments, examinations, and other learning activities. The Academic Skills and Learning Centre offers a number of workshops and seminars that you may find useful for your studies.
Online SubmissionThe ANU uses Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. While the use of Turnitin is not mandatory, the ANU highly recommends Turnitin is used by both teaching staff and students. For additional information regarding Turnitin please visit the ANU Online website.
Hardcopy SubmissionFor some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.
No submission of assessment tasks without an extension after the due date will be permitted. If an assessment task is not submitted by the due date, a mark of 0 will be awarded.
Referencing RequirementsAccepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.
Student work will be returned online or in assignment boxes.
Extensions and PenaltiesExtensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure The Course Convener may grant extensions for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.
Distribution of grades policyAcademic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes. Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.
Support for studentsThe University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).
- ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
- ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
- ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
- ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
- ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
- ANUSA supports and represents undergraduate and ANU College students
- PARSA supports and represents postgraduate and research students
Number theory and algebraic geometry. Also algebra more broadly and category theory.
Dr James Borger