# MATH3962 Rings, Fields and Galois Theory (Advanced)

## General Information

This page contains information on the Senior advanced Unit of Study MATH3962 Rings, Fields and Galois Theory (Advanced).

- 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.

Assignments Assessment Lecture notes References Timetable Tutorials

## MATH3962 Information in 2017

## 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, with references to the lecture notes by Bob Howlett [RH] and Adrian Nelson [AN], as well as the textbook [IS] by Ian Stewart. These references are made for your convenience only and you should be aware that the material covered in lectures, and the order in which it is presented, is not exactly the same as in these sets of lecture notes. The lectures are the definitive guide for the content of this course.

- Week 1
- Roadmap of the course, the integers, and a first discussion of rings.
- Week 2
- Some basic ring theory: homomorphisms, subrings, ideals, quotient rings, the First Isomorphism Theorem

See [RH, Chapters 2, 5-8] and [AN, Chapters 1-2]. - Week 3
- Correspondence Theorem. Integral domains. If R is an
integral domain so is \(R[x]\). Maximal and prime ideals, fields of fractions, classification of the
ideals of Z.

See [RH, Chapters 4, 6, 7] and [AN, Chapters 2, 4]. - Week 4
- Principal Ideal Domains (PIDs). If F is a field then F[x] is a PID, Gaussian integers. Euclidean domains, factorization, associates, greatest common divisors. Euclid's Lemma

See [RH, Chapters 9] and [AN, Chapters 3, 5]. - Week 5
- Unique Factorization Domains. Every PID is a UFD. Irreducible Gaussian integers, Fermat's 2 Square Theorem. Irreducible elements in the Hurwitz Quaternions.

See [RH, Chapter 9] and [AN, Chapter 5]. - Week 6
- If R is a UFD then so is R[X]. Factorising polynomials.

See [RH, Chapters 9] and [AN, Chapters 3, 5]. - Week 7
- Characteristic, Prime subfield, field extensions, algebraic and transcendental elements

See [RH, pp56-62] and [AN, 6.1-6.5] and [IS, Chapters 4-6, 16, 17.2] - Week 8
- Irreducible polynomials, Gauss's Lemma, Eisenstein's criterion, the Rational Roots Theorem

See [RH, pp54-56, 62-66] and [AN, 5.2-5.4] and [IS, Chapter 3] - Week 9
- Kronecker's algorithm, Berlekamp's algorithm, finite fields

See [IS, Chapter 20] - Week 10
- Splitting fields, normal extensions, separable extensions

See [RH, pp82-86] and [AN, 6.8-6.10, 7.7] and [IS, Chapter 9, 17.3-17.5] - Week 11
- Groups acting on fields, group characters, the Galois correspondence

See [RH, pp87-94] and [AN, 7.1-7.5] and [IS, Chapters 10-12, 17.6] - Week 12
- The Fundamental Theorem of Galois Theory, straight-edge and compass constructions

See [RH, pp95-105, 66-69] and [AN, 7.6, 6.6-6.7] and [IS, Chapters 22, 7, 19] - Week 13
- Revision

## 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 is due Thursday of week 6 (13th April): The questions are available here. Please read the instructions on the question sheet carefully.
- Assignment 2 will be due Thursday of week 11 (25th May), and will be posted here in due course.

- To help you study, here is a copy of last year's Assignment 1, and the solutions. Here is a copy of last year's Assignment 2, with 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 25th May (Week 11)

**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).

## 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

## 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.

- 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
- Lecture 16
- Lecture 17
- Lecture 18 For a short proof of the Prime Value Test (Theorem 18.6), see the very readable and freely available paper by M. Ram Murty, "Prime numbers and irreducible polynomials", in American Mathematical Monthly, 109, No 5, 2002, 452-458.
- 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

## 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:

- 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 will not be provided, however you can come and ask questions at the consultations.

## Timetable

Show timetable / Hide timetable.