• Class Number 2968
• Term Code 3330
• Class Info
• Unit Value 6 units
• Mode of Delivery In Person
• COURSE CONVENER
• Dr Anand Deopurkar
• LECTURER
• Dr Anand Deopurkar
• Class Dates
• Class Start Date 20/02/2023
• Class End Date 26/05/2023
• Census Date 31/03/2023
• Last Date to Enrol 27/02/2023
SELT Survey Results

Algebra 2: Field extensions and Galois Theory (MATH6215)

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.

Topics to be covered include:

• Galois Theory - fields
• Field extensions
• Normal extensions
• Separable extensions
• Revision of group theory, abelian and soluble groups
• Galois' Theorem
• Solubility of equations by radicals
• Finite fields
• Cyclotomic fields

Note: Graduate students attend joint classes with undergraduates but will be assessed separately.

## Learning Outcomes

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.

Recommended student system requirements

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

## Staff Feedback

Students will be given feedback in the following forms in this course:

• feedback to the whole class, to groups, to individuals.

## Student Feedback

ANU 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). Feedback can also be provided to Course Conveners and teachers via the Student Experience of Learning & Teaching (SELT) feedback program. SELT surveys are confidential and also provide the Colleges and ANU Executive with opportunities to recognise excellent teaching, and opportunities for improvement.

## Class Schedule

Week/Session Summary of Activities Assessment
1 Factorisation of integers and polynomials
2 Factorisation in more exotic rings Quiz 1
3 Algebraic and transcendental numbers Homework 1 due
4 Fields and field extensions Quiz 2
5 Application to geometry: ruler and compass constructions Homework 2 due
6 Creating new fields. Finite fields Quiz 3
7 Function fields. The fundamental theorem of algebra. Midterm exam
8 Splitting elds, symmetries of field extensions Homework 3 due
9 Main theorem of Galois theory Quiz 4
10 Applications to solving cubic equations Homework 4 due
11 Solvability by radicals Quiz 5
12 Additional topics Homework 5 due

## Tutorial Registration

Information about workshops, how and when they will be available will be given on the course Wattle site.

## Assessment Summary

Assessment task Value Due Date Return of assessment Learning Outcomes
Assignments 25 % * * 1,2,3,4
In class quizzes 25 % * * 1,2,3,4
Midterm examination 20 % * * 1,2,3,4
Final examination 20 % 01/06/2023 29/06/2023 1,2,3,4
Expository paper 10 % * * 1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

## Policies

ANU 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 Integrity Rule before the commencement of their course. Other key policies and guidelines include:

## Assessment Requirements

The 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 Academic Skills website. In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

## Moderation of Assessment

Marks 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.

## Participation

In Semester 1 2023, this course is delivered on campus.

## Examination(s)

This course includes a midterm and final examination. The details and mode of delivery for exams will be communicated through the course Wattle site and the ANU final examination timetable.

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 mode of the exam.

Value: 25 %
Learning Outcomes: 1,2,3,4

Assignments

There are 5 homework assignments due over the semester worth 25% total.

Value: 25 %
Learning Outcomes: 1,2,3,4

In class quizzes

On alternate Wednesdays, there will be an a short in-class quiz. It will include individual as well as group effort. See the Wattle site for details.

Value: 20 %
Learning Outcomes: 1,2,3,4

Midterm examination

In Week 7, there will be an in-person mid-term examination. See the Wattle site for more details.

Value: 20 %
Due Date: 01/06/2023
Return of Assessment: 29/06/2023
Learning Outcomes: 1,2,3,4

Final examination

he 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.

date, time and mode of the examination.

Value: 10 %
Learning Outcomes: 1,2,3,4

Expository paper

Students will write a 5-10 page account of a topic related to the course.

Academic integrity is a core part of the ANU culture as a community of scholars. The University’s students are an integral part of that community. The academic integrity principle commits all students to engage in academic work in ways that are consistent with, and actively support, academic integrity, and to uphold this commitment by behaving honestly, responsibly and ethically, and with respect and fairness, in scholarly practice.

The University expects all staff and students to be familiar with the academic integrity principle, the Academic Integrity Rule 2021, the Policy: Student Academic Integrity and Procedure: Student Academic Integrity, and to uphold high standards of academic integrity to ensure the quality and value of our qualifications.

The Academic Integrity Rule 2021 is a legal document that the University uses to promote academic integrity, and manage breaches of the academic integrity principle. The Policy and Procedure support the Rule by outlining overarching principles, responsibilities and processes. The Academic Integrity Rule 2021 commences on 1 December 2021 and applies to courses commencing on or after that date, as well as to research conduct occurring on or after that date. Prior to this, the Academic Misconduct Rule 2015 applies.

The University commits to assisting all students to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. All coursework students must complete the online Academic Integrity Module (Epigeum), and Higher Degree Research (HDR) students are required to complete research integrity training. The Academic Integrity website provides information about services available to assist students with their assignments, examinations and other learning activities, as well as understanding and upholding academic integrity.

## Online Submission

You will be required to agree to a declaration as part of the submission of your assignments, that will record your understanding of ANU academic integrity principles.  Please keep a copy of the assignment for your records. MATH6215 does not use Turnitin, having been granted an exemption.

## Hardcopy Submission

For 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.

## Late Submission

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. An extension request must be submitted 1 day (24 hours) before the due date. Extensions will be granted only in exceptional circumstances.

## Referencing Requirements

The Academic Skills website has information to assist you with your writing and assessments. The website includes information about Academic Integrity including referencing requirements for different disciplines. There is also information on Plagiarism and different ways to use source material.

## Returning Assignments

Student work will be returned online or in assignment boxes.

## Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted 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.

## Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Academic 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 students

The 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).

## Convener

 Dr Anand Deopurkar Anand.Deopurkar@anu.edu.au

### Research Interests

Algebraic geometry, representation theory, number theory, and related topics in algebra, geometry, and analysis.

### Dr Anand Deopurkar

 Monday 13:00 14:00 Wednesday 14:00 15:00

## Instructor

 Dr Anand Deopurkar 61252908 Anand.Deopurkar@anu.edu.au

### Dr Anand Deopurkar

 Monday 13:00 14:00 Wednesday 14:00 15:00