MATH32012 - Commutative Algebra
Year: 3 -
Semester: 2 -
Credit Rating: 10
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 (Grbner bases - an extension of the Euclidean division algorithm to polynomials in 2 or more variables), together with a selection of more classical material.
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.
On successful completion of this course unit students will be able to demonstrate
- facility in dealing with polynomials (in one and more variables);
- understanding of some basic ideal structure of polynomial rings;
- appreciation of the subtleties of factorisation into prime and irreducible elements;
- ability to compute generating sets and GrÃ¶bner bases for ideals in polynomial rings;
- ability to relate polynomials to other algebraic structures such as algebraic varieties;
- ability to solve problems relating to the factorisation of polynomials, irreducible polynomials and polynomial equations in several variables.
- Future topics requiring this course unit
None, though the material connects usefully with algebraic geometry and Galois theory.
1.Ideals in commutative rings: euclidean rings, principal ideal rings, noetherian rings. 
2.Ideals in polynomial rings: monomial orderings, Grbner bases, Hilbert's basis theorem. 
3.Computing ideals in polynomial rings: division algorithm, Buchberger's algorithm. 
4.Factorisation: irreducible and prime elements, unique factorisation domains, Gauss's Lemma, Eisenstein's criterion, fields of fractions. 
5.Zero sets of polynomials: algebraically closed fields, affine varieties, radical of an ideal, elimination method, the Nullstellensatz. 
Teaching & Learning Process (Hours Allocated To)
Assessment and Feedback
- Coursework: weighting 20%
- End of semester examination: two hours weighting 80%
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[1,2] are are useful general references on algebra though neither covers more advanced content of the course such as Grbner bases. For Grbner bases see . A treatment of factorisation as well as Grbner bases, with many exercises, is given in .  is a new textbook which combines Grbner 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.]