Undergraduate Course: Commutative Algebra (MATH10017)
|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
|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 non-commutative algebra.
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)
|| Students MUST have passed:
Honours Algebra (MATH10069)
||Other requirements|| None
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2015/16, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20%, Examination 80%
||Hours & Minutes
|Main Exam Diet S1 (December)||Commutative Algebra (MATH10017) ||2:00|
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.
|Cox, Little, O'Shea: Ideals, Varieties and Algorithms. An introduction to computational Algebraic Geometry and Commutative Algebra|
Reid: Undergraduate Commutative algebra
|Course organiser||Dr Milena Hering
|Course secretary||Mrs Alison Fairgrieve
Tel: (0131 6)50 5045
© Copyright 2015 The University of Edinburgh - 18 January 2016 4:24 am