# MATH3962/4062 Rings, Fields and Galois Theory (Advanced)

## General Information

This page contains information on the senior advanced unit of study MATH3962.

- Taught in Semester 1.
- Credit point value: 6.
- Classes per week: Three lectures and one tutorial.
- Lecturer(s): James Parkinson .

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the handbook entry for MATH3962 for further information relating to MATH3962.

You may also view the description of MATH3962 and the description of MATH4062 in the University's course search database.

For enrolled students or other authorized people only, here is a link to the Canvas page for MATH3962.

Students have the right to appeal any academic decision made by the School or Faculty: see sydney.edu.au/students/academic-appeals.html.

## MATH3962 Information in 2019

## Class and consultation times

- Lectures will be held on Mondays, Wednesdays and Thursdays at 14:00.
- My consultation time is Wednesday 1-2pm in Carslaw 614.

## Unit outline

This unit of study investigates the modern 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 tools needed for this program include the fundamental concepts of groups, rings, and fields. The course begins with the definitions and examples of these concepts, as well as the associated structures such as subgroups, subrings, homomorphisms, ideals and quotients. 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 links the process of solving polynomials with the process of taking field extensions, and then links this process to properties of the Galois group of the polynomial. Of course there is a lot to learn before understanding how this all works, but the point is that the problem of solving polynomial equations is converted into a much more accessible problem in finite group theory.

Along the way we 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.

Here is a week-by-week plan of the topics that we will cover. However things might change, and the lectures are the definitive guide for the content of this course.

- Week 1
- Introduction and overview, definitions of groups and rings, examples
- Week 2
- Subgroups, cosets, Lagrange's Theorem, normal subgroups, quotient groups, the symmetric group
- Week 3
- Subrings, polynomial rings, homomorphisms, ideals, and the First Isomorphism Theorem for groups and rings
- Week 4
- The Correspondence Theorem, integral domains, field of fractions of an integral domain
- Week 5
- Principal ideal domains, Euclidean domains, greatest common divisors, prime and irreducible elements
- Week 6
- The Unique Factorisation Theorem, unique factorisation domains, case study: Gaussian integers.
- Week 7
- Unique factorisation in polynomial rings, irreducibility in polynomial rings
- Week 8
- Irreducibility in polynomial rings continued, ring and field extensions
- Week 9
- Minimal polynomials, degree of a field extension, constructible numbers
- Week 10
- Solution to constructibility problems, constructible polygons, splitting fields, separability
- Week 11
- Finite fields, Galois groups, statement of the Galois correspondence, the order of the Galois group
- Week 12
- Proof of the Galois correspondence, solving polynomial equations using radicals, insolubility of the general quintic
- Week 13
- Revision and tying off loose ends.

## Learning outcomes

The learning outcomes for this unit of study are as follows.

- be familiar with the basics of abstract ring and field theory;
- be familiar with the concepts of integral domains, principal ideal domains, Euclidean domains, and unique factorisation domains, and understand the relationships between these concepts;
- understand the concept of irreducibility in integral domains;
- be proficient at applying various irreducibility tests;
- be proficient at applying the Euclidean Algorithm in various contexts;
- have a solid working knowledge of the basic examples of rings and fields including the integers, Gaussian integers, polynomial rings, the rational numbers, and finite fields;
- be able to work with field extensions, including computing the degree of an extension and the minimal polynomial of a simple extension;
- understand the solutions to the three ancient greek geometric problems;
- know and be able to apply the basic concepts and definitions from Galois Theory;
- know the basic properties of the Galois group of a field extension;
- be able to compute Galois groups in simple examples;
- be able to construct proofs, including sophisticated proofs using a variety of concepts covered in the unit;
- be proficient in dealing in abstract concepts with an emphasis on the clear explanation of such concepts to others;
- be able to apply the theory and methods introduced in the unit to specific examples, both those encountered in lectures and tutorials, and to related examples.

## Assessment

- Assignment 1 was due Thursday 4th April. The questions are available here. The solutions are available here. You can also find some general feedback here.
- Assignment 2 was due Thursday 23rd May. The questions are available here. The solutions are available here. You can also find some general feedback here.

Your mark for MATH3962 will be calculated as follows.

**Two assignments**, worth 10% each. The assignments will give practice in investigating examples and constructing proofs, and feedback should help with your mathematical writing skills and exam preparation. The assignments are due (via LMS) by midnight on the following dates:- Assignment 1 due on Thursday 4th April (Week 6)
- Assignment 2 due on Thursday 23rd May (Week 12)

**Tutorial participation**, worth 10%. The tutorials will require your active participation: mathematics is not a spectator sport. We will be working through the questions together. The tutorials are an integral part to the course, since the lectures are pretty dense and theory based. So it is absolutely essential that you attend. You will be awarded one mark per tutorial, up to a maximum of 10 marks, provided that you ACTIVELY participate in the tutorial.The tutorial sheets will be posted below. We won't get through all the questions in the tutorial. It is expected that you spend at least 3 or 4 hours of your own time each week finishing off as many of the questions as you can. This is key to success in this challenging course.

**Final exam**, 2 hours long and worth 70%, during examination period. No notes, books, or calculators are allowed (no questions will require calculators).

## Grade descriptors

- High Distinction (HD), 85-100
- Complete or close to complete mastery of the material
- Distinction (D), 75-84:
- Excellence, but substantially less than complete mastery
- Credit (CR), 65-74:
- A creditable performance that goes beyond routine knowledge and understanding, but less than excellence
- Pass (P), 50-64:
- At least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

## Reference books

The content of the unit is defined by the lectures rather than by a set text. Even though there is no reference book for the course, students might find the following lecture notes from previous years helpful:

- [RH] Rings and Fields and an Introduction to Galois Theory, by Robert Howlett,
- [AN] Rings, Fields and Galois Theory, by Adrian Nelson.

It is always a good idea to consult other sources for extra problems and alternative explanations. Most online mathematical encyclopedias contain material relevant to this unit. Be aware that conventions and notation may differ slightly from those in the lectures. The following books could be used to provide further practice if you like:

*Abstract Algebra*, D. Dummit and R. Foote (this is an excellent reference, also for group theory)*Galois theory*, E. Artin*A survey of modern algebra*, Garrett Birkhoff and Saunders Mac Lane*Modern algebra: an Introduction*, John R. Durbin*A first course in abstract algebra*, John B. Fraleigh*Abstract algebra*, I. N. Herstein- [IS]
*Galois theory*, I. N. Stewart

## Linear and Abstract Algebra Revision: Essential reading!

Below is a very useful summary of what you covered in MATH2922. It is absolutely essential that you have a good grasp of this material. In particular, you have to know all of the material on groups, fields, vector spaces, linear transformations, and matrix representations. This is the material up to and including Lecture 5-3 in the notes:

In particular, it is important that you have a good understanding of group theory. Here are some questions on group theory and here are the solutions.

## Lecture material

The following lecture notes are close approximations to what was covered in lectures. Some proofs and/or details that were skipped in lectures might be contained in these notes (otherwise they are exercises). The links will become active as the semester progresses.

- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5
- Lecture 6
- Lecture 7
- Lecture 8
- Lecture 9
- Lecture 10
- Lecture 11
- Lecture 12
- Lecture 13
- Lecture 14
- Lecture 15 and some optional supplementary material on the quaternions and Lagrange's 4 Square Theorem (for interest only).
- Lecture 16
- Lecture 17
- Lecture 18
- Lecture 19
- Lecture 20
- Lecture 21
- Lecture 22
- Lecture 23
- Lecture 24
- Lecture 25
- Lecture 26
- Lecture 27
- Lecture 28
- Lecture 29
- Lecture 30
- Lecture 31
- Lecture 32
- Lecture 33
- Lecture 34
- Lecture 35
- Lecture 36
- Lecture 37 and 38

## Tutorial questions and solutions

Tutorials will be held in Weeks 2-13 (so the first tutorial is in week 2). Please attend the tutorial on your timetable.

Tutorials questions and solutions can be downloaded below. All question sheets are active links, however the links to the solutions will only become active as the semester progresses:

- Tutorial Week 2 and solutions
- Tutorial Week 3 and solutions
- Tutorial Week 4 and solutions
- Tutorial Week 5 and solutions
- Tutorial Week 6 and solutions
- Tutorial Week 7 and solutions
- Tutorial Week 8 and solutions
- Tutorial Week 9 and solutions
- Tutorial Week 10 and solutions
- Tutorial Week 11 and solutions
- Tutorial Week 12 and solutions
- Tutorial Week 13 and solutions

**Math3962 exam**

The Math3962 end of semester exam will consist of five questions, each of which may be (and should be), attempted.

Here is a selection of past exams (the more recent papers are perhaps more relevant in terms of content):

Solutions to the 2017 exam are here. Further solutions will not be provided (however, see week 13 lectures for 2018 exam solutions). You can come and ask questions at the exam consultation during stuvac:

## Timetable

Show timetable / Hide timetable.