BSc Actuarial Science and Mathematics

Year of entry: 2022

Course unit details:
Commutative Algebra

Unit code MATH32012
Credit rating 10
Unit level Level 3
Teaching period(s) Semester 2
Offered by Department of Mathematics
Available as a free choice unit? No

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.

Pre/co-requisites

Unit title Unit code Requirement type Description
Algebraic Structures 2 MATH20212 Pre-Requisite Compulsory
Algebraic Structures 1 MATH20201 Pre-Requisite Compulsory

Aims

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.

 

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]

Assessment methods

Method Weight
Other 20%
Written exam 80%
  • Coursework: weighting 20%
  • End of semester examination: weighting 80%

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.

Recommended reading

Core:
 D.A. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms (3rd edition), Springer 2007.
Essential:
Reid, Miles. Undergraduate Commutative Algebra: London Mathematical Society Student Texts. Cambridge, UK: Cambridge University Press, April 26, 1996. 
Further reading:
Atiyah, Michael, and Ian Macdonald. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1994.
Eisenbud, David. Commutative Algebra: With a View Toward Algebraic Geometry. New York, NY: Springer-Verlag, 1999.

Study hours

Scheduled activity hours
Lectures 12
Tutorials 12
Independent study hours
Independent study 76

Teaching staff

Staff member Role
Martin Orr Unit coordinator

Additional notes

The independent study hours will normally comprise the following. During each week of the taught part of the semester:

·         You will normally have approximately 60-75 minutes of video content. Normally you would spend approximately 2-2.5 hrs per week studying this content independently

·         You will normally have exercise or problem sheets, on which you might spend approximately 1.5hrs per week

·         There may be other tasks assigned to you on Blackboard, for example short quizzes or short-answer formative exercises

·         In some weeks you may be preparing coursework or revising for mid-semester tests

Together with the timetabled classes, you should be spending approximately 6 hours per week on this course unit.

The remaining independent study time comprises revision for and taking the end-of-semester assessment.

The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.

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