# MATH3962 Rings, Fields and Galois Theory (Advanced)

## General Information

• 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 MATH3962 handbook entry for further information relating to MATH3962.

You may also view the description of MATH3962 in the central units of study database.

Students have the right to appeal any academic decision made by the School or Faculty. For further information, see the Science Faculty web site.

## Class and consultation times

• Lectures will be held on Mondays, Wednesdays and Thursdays at 14:00 in Carslaw 375.
• Tutorials will be held on Tuesdays in Weeks 2-13 at 10:00 in Carslaw 350.
• My consultation time is Monday 1-2pm in Carslaw 614.

## Unit outline

This unit has three main aims:

• To advance your mathematical education by introducing the language and basic constructions of ring theory which are encountered in most branches of modern mathematics.
• To look in detail at the theory of fields as applied to one of the earliest motivational problems of algebra, solving polynomial equations.
• To develop one of the most beautiful gems of mathematics, the Galois Theory of polynomial equations, to the extent that we can answer and understand why is there no general formula giving the roots of a general polynomial of degree five or higher, in terms of its coefficients using only the basic algebraic operations of addition, subtraction, multiplication, division and forming radicals – that is, square roots, cube roots etc.

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

Week 1
Introduction and overview, the ring of integers, definition of rings and fields
Week 2
Subrings, polynomial rings, homomorphisms, ideals, and the First Isomorphism Theorem
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
Mid-semester review, unique factorisation in polynomial rings, irreducibility in polynomial rings
Week 7
Irreducibility in polynomial rings continued, Kronecker's algorithm, ring and field extensions
Week 8
Field extensions continued, minimal polynomials, degree of a field extension, constructible numbers
Week 9
Solution to constructibility problems, constructible polygons, splitting fields, separability
Week 10
Finite fields, Galois groups, statement of the Galois correspondence
Week 11
The order of the Galois group, proof of the Galois correspondence
Week 12
Solving polynomial equations using radicals, insolubility of the general quintic
Week 13
Revision and tying off loose ends
This is subject to change, depending on our progress and inspiration.

## Learning outcomes

By the end of the unit, you should be:

• Proficient in dealing in abstract concepts with an emphasis on the clear explanation of such concepts to others;
• 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 of week 6 (13th April): The questions are available here and solutions.
• Assignment 2 was Thursday of week 12 (1st June). The questions are available here and solutions.

Your mark for MATH3962 will be calculated as follows.

• Two assigments, 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 at the start of the lectures on the following dates:
• Assignment 1 due on Thursday 13th April (Week 6)
• Assignment 2 due on Thursday 1st June (Week 12)
The assignments will be available for download from this web page later in the semester
• Tutorial participation, worth 10%. The tutorials will be discussion based. 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.

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 all the questions. 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).

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:

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
• [ISGalois theory, I. N. Stewart

## Lecture Notes

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.

## Tutorial questions and solutions

Tutorials will be held on Tuesdays in Weeks 2-13 at 10:00 in Carslaw 350.
The first tutorial is in week 2.

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

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):

2016     2014     2013     2011     2009

Solutions will not be provided, however you can come and ask questions at the consultations:

• Tuesday 20th June 1-3pm (Carslaw 614)
• Tuesday 27th June 1-3pm (Carslaw 614)
• ## Timetable

Show timetable / Hide timetable.