Unit outline_

# MATH4062: Rings, Fields and Galois Theory

## Overview

This unit of study lies at the heart of modern algebra. In the unit we investigate the mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a large enough field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalises the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions. Along the way you will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient Greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the quadratic formula for the general quintic equation. On completing this unit of study you will have obtained a deep understanding of modern abstract algebra.

### Unit details and rules

Academic unit Mathematics and Statistics Academic Operations 6 (MATH2922 or MATH2961) or a mark of 65 or greater in (MATH2022 or MATH2061) or 12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979) None MATH3062 or MATH3962 None Yes

### Teaching staff

Coordinator Kevin Coulembier, kevin.coulembier@sydney.edu.au

## Assessment

Type Description Weight Due Length
Final exam (Take-home short release) Final exam
Written exam - Week 17
60% Formal exam period 2 hours
Outcomes assessed:
Assignment Assignment 1
Typed mathematical assignment
20% Week 07
Due date: 05 Apr 2022 at 21:00

Closing date: 12 Apr 2022
3-5 extended mathematical questions
Outcomes assessed:
Assignment Assignment 2
Typed mathematical assignment
20% Week 12
Due date: 17 May 2022 at 22:00

Closing date: 24 May 2022
3-5 extended mathematical questions
Outcomes assessed:
= Type D final exam

### Assessment summary

• Written assignments: These will consist of 3-4 extended mathematical questions. Students will be required to type their solutions (a template will be provided). They will need to create a consice written argument.
• Final exam: The exam will cover all material in the unit from both lectures and tutorials. The exam will be all extended answer questions. If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinato

Detailed information for each assessment can be found on Canvas

### Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Complete or close to complete mastery of the material.

Distinction

75 - 84

Excellence, but substantially less than complete mastery.

Credit

65 - 74

A creditable performance that goes beyond routine knowledge and understanding, but less than excellence.

Pass

50 - 64

At least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Late submissions are not accepted without special consideration.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

## Learning support

### Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

### Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

### Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Introduction and overview, definitions of groups and rings, examples Lecture and tutorial (4 hr)
Week 02 Subgroups, cosets, Lagrange's Theorem, normal subgroups, quotient groups, the symmetric group Lecture and tutorial (4 hr)
Week 03 Subrings, polynomial rings, homomorphisms, ideals, and the First Isomorphism Theorem for groups and rings Lecture and tutorial (4 hr)
Week 04 The correspondence theorem, integral domains, field of fractions of an integral domain Lecture and tutorial (4 hr)
Week 05 Principal ideal domains, Euclidean domains, greatest common divisors, prime and irreducible elements Lecture and tutorial (4 hr)
Week 06 The unique factorisation theorem, unique factorisation domains, case study: Gaussian integers Lecture and tutorial (4 hr)
Week 07 Unique factorisation in polynomial rings, irreducibility in polynomial rings Lecture and tutorial (4 hr)
Week 08 Irreducibility in polynomial rings continued, ring and field extensions Lecture and tutorial (4 hr)
Week 09 Minimal polynomials, degree of a field extension, constructible numbers Lecture and tutorial (4 hr)
Week 10 Solution to constructibility problems, constructible polygons, splitting fields, separability Lecture and tutorial (4 hr)
Week 11 Finite fields, Galois groups, statement of the Galois correspondence, the order of the Galois group Lecture and tutorial (4 hr)
Week 12 Proof of the Galois correspondence, solving polynomial equations using radicals, insolubility of the general quintic Lecture and tutorial (4 hr)
Week 13 Revision and tying off loose ends Lecture and tutorial (4 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Recall basics of abstract ring and field theory
• LO2. Recall the concepts of integral domains, principal ideal domains, Euclidean domains, and unique factorisation domains, and explain the relationships between these concepts
• LO3. Determine irreducibility in an integral domain both by applying appropriate tests and by developing novel techniques and methods
• LO4. Calculate the greatest common divisor in various Euclidean domains via the Euclidean Algorithm
• LO5. Analyse foundational examples of rings and fields including the integers, Gaussian integers, polynomial rings, the rational numbers, and finite fields
• LO6. Compute the degree of an extension and analyse the minimal polynomial of a simple field extension
• LO7. Analyse and solve complex problems using the solutions to the three ancient Greek geometric problems
• LO8. Recall and apply the foundational concepts and definitions from Galois Theory
• LO9. Determine Galois groups in various examples
• LO10. Construct rigorous proofs, and manipulate and apply abstract concepts with an emphasis on the clear explanation of such concepts to others

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.

### Outcome map

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

No changes have been made since this unit was last offered.

### Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

General Laboratory Safety Rules

• No eating or drinking is allowed in any laboratory under any circumstances
• A laboratory coat and closed-toe shoes are mandatory