## Commutative Algebra

 Unit code: MATH32011 Credit Rating: 10 Unit level: Level 3 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

#### Aims

The course unit will deepen and extend students' knowledge and understanding of commutative algebra. By the end of the course unit the student will have learned more about familiar mathematical objects such as polynomials and algebraic numbers, will have acquired various computational and algebraic skills and will have seen how the introduction of structural ideas leads to the solution of mathematical problems.

#### Overview

The central theme of this course is factorisation (theory and practice) in commutative rings; rings of polynomials are our main examples but there are others, such as rings of algebraic integers.

Polynomials are familiar objects which play a part in virtually every branch of mathematics. Historically, the study of solutions of polynomial equations (algebraic geometry and number theory) and the study of symmetries of polynomials (invariant theory) were a major source of inspiration for the vast expansion of algebra in the 19th and 20th centuries.

In this course the algebra of polynomials in n variables over a field of coefficients is the basic object of study. The course covers fairly recent advances which have important applications to computer algebra and computational algebraic geometry (Gröbner bases - an extension of the Euclidean division algorithm to polynomials in 2 or more variables), together with a selection of more classical material.

#### Learning outcomes

On successful completion of this course unit students will be able to demonstrate:

• define irreducible and prime elements of commutative rings and calculate the groups of units of some rings;
• define what is meant by an Euclidean domain and calculate Euclidean functions for some rings such as Gaussian integers;
• use Eisenstein's criterion to determine whether a given polynomial is irreducible;
• use Gauss' lemma to prove that polynomial rings in several variables are unique factorisation domains;
• state Hilbert's theorems on ideals of polynomial rings in several variables and use them to relate polynomials to algebraic varieties;
• define Gröbner bases of ideals in polynomial rings and use them to calculate generating sets of some ideals in polynomial rings in two or three variables.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

• Coursework: weighting 20%
• End of semester examination: two hours weighting 80%

#### Syllabus

1.Ideals in commutative rings: euclidean rings, principal ideal rings, noetherian rings. [4]

2.Ideals in polynomial rings: monomial orderings, Gröbner  bases, Hilbert's basis theorem. [4]

3.Computing ideals in polynomial rings: division algorithm, Buchberger's algorithm. [4]

4.Factorisation: irreducible and prime elements, unique factorisation domains, Gauss's Lemma, Eisenstein's criterion, fields of fractions. [6]

5.Zero sets of polynomials: algebraically closed fields, affine varieties, radical of an ideal, elimination method, the Nullstellensatz. [4]

[1,2] are are useful general references on algebra though neither covers more advanced content of the course such as Gröbner bases. For Gröbner bases see [3]. A treatment of factorisation as well as Gröbner bases, with many exercises, is given in [4]. [5] is a new textbook which combines Gröbner bases with more traditional material and contains some practical examples.

1.R.B.J.T. Allenby, Rings, Fields and Groups: An Introduction to Abstract Algebra, Edward Arnold, 0-3405-4440-6.

2.J.B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley,1994, 0-201-59291-6.

3.D.A. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms (3rd edition), Springer 2007, 978-0387356501.

4.D.A. Dummit, R.M. Foote, Abstract Algebra (3rd edition), Wiley & Sons 2003, 978-0471433347. [Chapters 7, 8, 9 and 15.]

5.N. Lauritzen, Concrete Abstract Algebra, Cambridge University Press 2003, 978-0521534109. [Chapters 3, 4 and 5.]

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 22 hours
• Tutorials - 11 hours
• Independent study hours - 67 hours

#### Teaching staff

Alexander Premet - Unit coordinator