Commutative Algebra
| Code | School | Level | Credits | Semesters |
| MATH4087 | Mathematical Sciences | 4 | 20 | Spring UK |
- Code
- MATH4087
- School
- Mathematical Sciences
- Level
- 4
- Credits
- 20
- Semesters
- Spring UK
Summary
Commutative algebra is the foundation of modern pure mathematics: everything from Fermat’s Last Theorem to cryptography depends on the ideas developed in Commutative Algebra. Indeed, the solutions to many questions can be described in terms of systems of polynomial equations – think of this as a generalisation of linear algebra. Such systems of equations are also used in many applications, like in robotics, architecture, or chemistry. For linear equations you would find the solution using a matrix, but what do we do for polynomial equations?
This course presents and discusses commutative ring theory, the language of modern mathematics. Properties of polynomials and systems of polynomial equations are investigated. The course further introduces the most widely used approach to solve such systems: Gröbner bases, which can be thought of as analogous to computing the Reduced Row Echelon Form for a matrix. This allows a hands-on discussion of algebraic varieties, and their associated algebraic structures.
Target Students
Single and Joint Honours students from the School of Mathematical Sciences. Mathematical Physics students.
Classes
One two-hour class and two one-hour classes per week timetabled centrally, some of which may be used for lectures or workshops.
Assessment
- 30% Coursework 1: Summative assessment based on in-class assessments that are distributed over the semester
- 70% Exam 1 (3-hour)
Assessed by end of spring semester
Educational Aims
As an introduction to one of the highlights of 20th century mathematics, commutative algebra provides the foundational language of modern pure mathematics. It draws inspiration from other areas such as geometry, topology, and analysis, elements of which are taught at previous levels, and embeds these methods in a unifying framework, namely algebra, whose basics have already been encountered in MATH2015. It develops concrete facility in the manipulation and study of algebraic systems, and thus provides an excellent hands-on entrance to more abstract modules on Algebraic Geometry or Number Theory. The course can be taken both as a preparatory course or in conjunction with these more abstract courses at the school.Learning Outcomes
L1- define and apply the main algebraic concepts related to polynomial rings;
L2- investigate basic properties of systems of polynomial equations;
L3- use and apply Gröbner basis methods to study systems of polynomial equations
L4- compare and contrast algebraic features of systems of polynomial equations;
L5- prove basic statements about polynomial rings including topics such as monomial ideals, Hilbert Basis Theorem, properties of Gröbner bases, and elimination and extension theorems.