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. You will learn practical methods for solving systems of polynomial equations, as well as important theoretical results, for example, Hilbert's basis theorem. 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, Lie theory, and noncommutative algebra. 
Course description 
In year 1 you learnt to solve systems of linear equations in many variables. But what about equations of higher degree?
You probably have encountered a few methods so far to find the zeroes of univariate polynomials.
But what about solutions to sets of polynomial equations in several variables? Such sets equations come up naturally  in kinematics, robotics, physics, statistics, biology, optimization, etc. In the first part of the course we will learn Buchberger's algorithm which finds the zeroes of systems of polynomial equations.
We will then move on to explore properties of the polynomial ring  for example, Hilbert's basis theorem that says that every ideal in the polynomial ring is finitely generated  as well as more general concepts in commutative algebra. There is also a close relationship to geometry in this class: 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

Academic year 2015/16, Available to all students (SV1)

Quota: None 
Course Start 
Semester 1 
Timetable 
Timetable 
Learning and Teaching activities (Further Info) 
Total Hours:
100
(
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
69 )

Assessment (Further Info) 
Written Exam
80 %,
Coursework
20 %,
Practical Exam
0 %

Additional Information (Assessment) 
Coursework 20%, Examination 80% 
Feedback 
Not entered 
Exam Information 
Exam Diet 
Paper Name 
Hours & Minutes 

Main Exam Diet S1 (December)  Commutative Algebra (MATH10017)  2:00  
Learning Outcomes
On completion of this course, the student will be able to:
 Gain familiarity with the polynomial ring and be able to perform basic operations with both elements and ideals.
 Use computational tools, especially Gröbner 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 be able to explain the proofs of the main results in the class without access to notes or other resources.
 Be able to 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 
Contacts
Course organiser  Dr Milena Hering
Tel:
Email: M.Hering@ed.ac.uk 
Course secretary  Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
Email: Alison.Fairgrieve@ed.ac.uk 

