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Semester 1

Commutative Algebra (MATH10017)







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Course 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 non-commutative 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.

Assessment Information

Written Exam 80%, Coursework 20%, Practical Exam 0%

Additional Assessment Information

Coursework 20%, Examination 80%

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