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): Kevin Coulembier .
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.
For enrolled students or other authorized people only, here is a link to the Canvas page for MATH4062.
Students have the right to appeal any academic decision made by the School or Faculty: see sydney.edu.au/students/academicappeals.html.
Assessment
Date^{*}  Description  Better mark  Weighting 

23:59 April 19  Assignment 1  20%  
23:59 May 24  Assignment 2  20%  
TBC  Exam  60% 
MATH3962 Information in 2020
Class and consultation times

Lectures will be held on Mondays, Wednesdays and Thursdays at 9am, via Zoom.
 Tutorials are on Wednesday at 3pm on campus in Carslaw and at 4pm online, starting week 2.
 My consultation time is Wednesday 23pm in Carslaw 717.
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 weekbyweek 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
 Subrings, polynomial rings, homomorphisms, ideals, and the First Isomorphism Theorem for groups and rings
 Week 3
 The Correspondence Theorem, integral domains, field of fractions of an integral domain
 Week 4
 Principal ideal domains, Euclidean domains, greatest common divisors, prime and irreducible elements
 Week 5
 The Unique Factorisation Theorem, unique factorisation domains, case study: Gaussian integers.
 Week 6
 Unique factorisation in polynomial rings, irreducibility in polynomial rings
 Week 7
 Irreducibility in polynomial rings continued, ring and field extensions
 Week 8
 Minimal polynomials, degree of a field extension, constructible numbers
 Week 9
 Solution to constructibility problems, constructible polygons, splitting fields, separability
 Week 10
 Subgroups, cosets, Lagrange's Theorem, normal subgroups, quotient groups, the symmetric group
 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
Your mark for MATH3962 will be calculated as follows (note that adjustments have been made as a result of moving online in response to covid19).
 Two assignments, worth
15% 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 Friday 16th April (Week 6)
 Assignment 2 due on Friday 21th May (Week 11)
 Final exam, worth 70%, during examination period.
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.
Grade descriptors
 High Distinction (HD), 85100
 Complete or close to complete mastery of the material
 Distinction (D), 7584:
 Excellence, but substantially less than complete mastery
 Credit (CR), 6574:
 A creditable performance that goes beyond routine knowledge and understanding, but less than excellence
 Pass (P), 5064:
 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 53 in the notes:
In particular, it is important that you have a good understanding of group theory. On canvas you will find the tutorial for week 1; there is no actual tutorial in week 1, but the tutorial sheet contains some questions for revision of group theory.
Online resources
Monday lectures  Tuesday lectures  Wednesday lectures  Tutorials  Assessment  

Week 1 1/35/3 
Lecture 11  Lecture 12  Lecture 12  
Week 2 8/312/3 
Lecture 21  Lecture 22  Lecture 22  Tutorial 2 questions Tutorial 2 solutions 

Week 3 15/319/3 
Lecture 31  Lecture 32  Lecture 32  Tutorial 3 questions Tutorial 3 solutions 

Week 4 22/326/3 
Lecture 41  Lecture 42  Lecture 42  Tutorial 4 questions Tutorial 4 solutions 

Week 5 29/32/4 
Lecture 51  Lecture 52  Lecture 52  Tutorial 5 questions Tutorial 5 solutions 

Midsemester break  
Week 6 12/416/4 
Lecture 61  Lecture 62  Lecture 62  Tutorial 6 questions Tutorial 6 solutions 

Week 7 19/423/4 
Lecture 71  Lecture 72  Lecture 72  Tutorial 7 questions Tutorial 7 solutions 
Assignment 1 (20%) Due 23:59 April 19 
Week 8 26/430/4 
Lecture 81  Lecture 82  Lecture 82  Tutorial 8 questions Tutorial 8 solutions 

Week 9 3/57/5 
Lecture 91  Lecture 92  Lecture 92  Tutorial 9 questions Tutorial 9 solutions 
Assignment 1  solutions 
Week 10 10/514/5 
Lecture 101  Lecture 102  Lecture 102  Tutorial 10 questions Tutorial 10 solutions 
Assignment 2  questions 
Week 11 17/521/5 
Lecture 111  Lecture 112  Lecture 112  Tutorial 11 questions Tutorial 11 solutions 

Week 12 24/528/5 
Lecture 121  Lecture 122  Lecture 122  Tutorial 12 questions Tutorial 12 solutions 
Assignment 2 (20%) Due 23:59 May 24 
Week 13 31/54/6 
Lecture 131  Lecture 132  Lecture 132  Tutorial 13 questions Tutorial 13 solutions 