Undergraduate Course: Commutative Algebra (MATH10017)
Course Outline
School  School of Mathematics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 10 (Year 4 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  This course will be an introduction to commutative algebra, mainly focusing on methods to work with polynomial rings. The course begins with the abstract foundations of commutative algebra including the Noetherian and Hilbert's basis theorem. After the foundations are established the course focuses on practical methods for solving systems of polynomial equations. An important branch of algebra in its own right, commutative algebra is an essential tool to explore several other areas of mathematics, such as algebraic geometry, number theory, Galois theory, Lie theory, and noncommutative algebra. 
Course description 
The course begins by reviewing and building upon the elementary ring theory seen Honours Algebra. The material will include (but is not limited to) integral domains, unique factorisation domains, Noetherian rings, Hilbert's basis theorem, prime and maximal ideals. This builds the necessary foundations for the second part of course and allows for transparent connections with other areas of mathematics.
The course then moves to develop methods to solve systems of polynomial equations. In linear algebra you learnt to solve systems of linear equations in many variables. You probably have encountered a few methods so far to find the zeroes of univariate polynomials. in this course we consider the more general case of systems of polynomial equations with many variables and arbitrary degree. Such sets equations come up naturally  in kinematics, robotics, physics, statistics, biology, optimization, etc. A key tool in the solution to this problem is Buchberger's algorithm and Groebner bases.
There is a close relationship to geometry in this class: we will discuss Hilbert's nullstellensatz which shows how solution sets to polynomial equations are the building blocks of algebraic varieties, the objects studied in algebraic geometry. This class will provide some concrete examples of the concepts you have learnt in Honours Algebra and give you tools to do computations with them.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 
Students MUST have passed:
Honours Algebra (MATH10069)

Corequisites  
Prohibited Combinations  
Other requirements  None 
Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information
Not being delivered 
Learning Outcomes
On completion of this course, the student will be able to:
 Gain familiarity with commutative rings and perform basic operations with both elements and ideals.
 Use computational tools, especially Groebner bases and the Buchberger algorithm, to solve problems in polynomial rings; for example the ideal membership problem, or finding solutions to polynomial equations, but also to be able to apply these tools without prompting.
 State accurately and explain the proofs of the main results in the class without access to notes or other resources.
 Produce examples illustrating the mathematical concepts learnt in the class.

Reading List
Cox, Little, O'Shea: Ideals, Varieties and Algorithms. An introduction to computational Algebraic Geometry and Commutative Algebra
Reid: Undergraduate Commutative algebra 
Additional Information
Course URL 
http://student.maths.ed.ac.uk 
Graduate Attributes and Skills 
Not entered 
Keywords  CoA,Polynomials,Algebra,Rings,Hilbert,Groebner basis,UFD,PID,Noetherian 
Contacts
Course organiser  Dr David Quinn
Tel:
Email: D.Quinn@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

