MATH32012 Commutative Algebra - 2012/13, Semester 2

Online Test

The Online Test is currently accessible via the MATH32012 course content page in Blackboard. You may retake the test for revision purposes (e.g., to practise the computation of Gröbner bases). It will not affect your coursework mark. (The coursework marks have been finalised and are available via the Grade Centre in Blackboard.)

Revision materials

The following materials are available: Lecture notes (single file, lectures only) * Revision questions * ANSWERS to revision questions

Session notes (most recent first)

Week 11, examples class * Week 11, revision lecture (slides) * Week 11, lecture 1

Week 01, example sheet * Week 01, examples class * Week 01, lecture 2 * Week 01, lecture 1

Module description and prerequisites

• You should have general facility for dealing with algebraic structures: complex numbers, sets, groups, rings, fields. For this reason, MATH20212 Algebraic Structures 2 is a prerequisite.

Many find MATH32012 Commutative Algebra the most advanced abstract algebra course they take as part of their degree. Nevertheless, the content of the course is not just a sequence of theorems and proofs. You are expected to learn methods of algebraic computation relating to polynomials in several variables.

Solving equations has been a driving force of algebra at least since the Babylonians learned to solve quadratic equations some 3700 years ago. The subject matter of this course is, however, informed by more recent developments.

The work of Hilbert in late 19th - 20th century was key to the modern treatment of multivariate polynomials and provided a basis for commutative algebra and algebraic geometry. His result that every (consistent) system of polynomial equations over an algebraically closed field has at least one solution is known as the Nullstellensatz. But an efficient method of finding such solutions by elimination was not found until 1965, when Buchberger invented Gröbner bases.

In the course, key theorems about the ring of polynomials in several variables will be rigorously proved. Algorithms relating to polynomials will be explained and supported by examples. This includes factorising polynomials into irreducible factors and computing a Gröbner basis of an ideal.

Results and methods of Commutative Algebra have applications in various branches of mathematics and computer science. Here are some puzzles which we may use in the course as an illustration for the main content. You are welcome to have a go at solving them!

• Question 1 (Fermat, 17th century). Find all integers a "sandwiched" between a square and a cube.

• Question 2. How many ways are there of placing 8 queens on a chessboard so that no two queens attack each other? What about n queens on an n×n chessboard?
• Question 3. How many distinct Sudoku boards are there? (A Sudoku board is a 9×9 square with a number from 1 to 9 in each cell, satisfying the Sudoku constraints.)

images from Wikimedia commons

Coursework

There will be 2 pieces of assessed coursework:
• Assessed homework 1 (see a link above): a take-home problem sheet set on Wednesday 27 February (week 5), due on Tuesday 12 March (week 7) at 4pm.
• Blackboard-based online test: a timed, open-book test which the students complete online; multiple attempts are allowed

Previous years' exams

Commutative algebra exam papers from years 2008-2012 are available here.

Arrangements

Timetable:
Tuesday 1pm-1:50pm, in Schuster Blackett; Thursday 3pm-3:50pm and 4pm-4:50pm, in Ellen Wilkinson C5.1
Lecturer:
Dr Yuri Bazlov
e-mail:
yuri.bazlov, append AT and manchester.ac.uk
office:
2.220 Alan Turing building
office hours:
Tuesday 2:30-3:30pm. I intend to be available in my office during the office hours, but students may come to see me at other times as well, or make an appointment by email.

Extras

What is a Gröbner Basis?, a short expository note by Bernd Sturmfels