• Class Number 3048
• Term Code 3230
• Class Info
• Unit Value 6 units
• Mode of Delivery In Person
• COURSE CONVENER
• AsPr James Borger
• LECTURER
• AsPr James Borger
• Class Dates
• Class Start Date 21/02/2022
• Class End Date 27/05/2022
• Census Date 31/03/2022
• Last Date to Enrol 28/02/2022
SELT Survey Results

Advanced Algebra 2: Field extensions and Galois Theory (MATH3345)

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.

## 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). 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.

## Class Schedule

Week/Session Summary of Activities Assessment
1 Symmetric functions
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

## Tutorial Registration

Information about workshops, how and when they will be available will be given on the course Wattle site. Remote participation options will be provided for students who require them due to travel restrictions or COVID-safe guidelines. However, it is possible that not all times will be available for both remote and in-person attendance. Please refer to the course Wattle site for more information.

## Assessment Summary

Assessment task Value Due Date Return of assessment Learning Outcomes
Assignments 30 % * * 1,2,3,4
Mid-semester exam 35 % * * 1,2,3,4
Final examination 35 % 02/06/2022 30/06/2022 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 Misconduct 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 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 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 2022, this course is delivered on campus with adjustments for remote participants. Remote students will be able to attend workshops virtually. They will also have access to Echo360 recordings of the lectures or video recordings of lectures from previous years.

## Examination(s)

This course includes a 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 time frame 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: 30 %
Learning Outcomes: 1,2,3,4

Assignments

There are 6 assignments due over the semester worth 5% 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.

Value: 35 %
Learning Outcomes: 1,2,3,4

Mid-semester exam

Please check the course Wattle site and the ANU Examination Timetable to confirm the date, time and mode of the mid semester exam.

Value: 35 %
Due Date: 02/06/2022
Return of Assessment: 30/06/2022
Learning Outcomes: 1,2,3,4

Final examination

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 mode of the examination.

## 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. MATH3345 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.

## Referencing Requirements

Accepted 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.

## 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 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.

## Privacy Notice

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

 AsPr James Borger James.Borger@anu.edu.au

### Research Interests

Number theory and algebraic geometry. Also algebra more broadly and category theory.

## Instructor

 AsPr James Borger 53750 James.Borger@anu.edu.au