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Unit of study_

MATH4062: Rings, Fields and Galois Theory

This unit of study lies at the heart of modern algebra. In the unit we investigate the mathematical theory that was originally developed for the purpose of studying polynomial equations. In a nutshell, the philosophy is that it should be possible to completely factorise any polynomial into a product of linear factors by working over a large enough field (such as the field of all complex numbers). Viewed like this, the problem of solving polynomial equations leads naturally to the problem of understanding extensions of fields. This in turn leads into the area of mathematics known as Galois theory. The basic theoretical tool needed for this program is the concept of a ring, which generalises the concept of a field. The course begins with examples of rings, and associated concepts such as subrings, ring homomorphisms, ideals and quotient rings. These tools are then applied to study quotient rings of polynomial rings. The final part of the course deals with the basics of Galois theory, which gives a way of understanding field extensions. Along the way you will see some beautiful gems of mathematics, including Fermat's Theorem on primes expressible as a sum of two squares, solutions to the ancient Greek problems of trisecting the angle, squaring the circle, and doubling the cube, and the crown of the course: Galois' proof that there is no analogue of the quadratic formula for the general quintic equation. On completing this unit of study you will have obtained a deep understanding of modern abstract algebra.

Code MATH4062
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
(MATH2922 or MATH2961) or a mark of 65 or greater in (MATH2022 or MATH2061) or 12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)
MATH3062 or MATH3962

At the completion of this unit, you should be able to:

  • LO1. Recall basics of abstract ring and field theory
  • LO2. Recall the concepts of integral domains, principal ideal domains, Euclidean domains, and unique factorisation domains, and explain the relationships between these concepts
  • LO3. Determine irreducibility in an integral domain both by applying appropriate tests and by developing novel techniques and methods
  • LO4. Calculate the greatest common divisor in various Euclidean domains via the Euclidean Algorithm
  • LO5. Analyse foundational examples of rings and fields including the integers, Gaussian integers, polynomial rings, the rational numbers, and finite fields
  • LO6. Compute the degree of an extension and analyse the minimal polynomial of a simple field extension
  • LO7. Analyse and solve complex problems using the solutions to the three ancient Greek geometric problems
  • LO8. Recall and apply the foundational concepts and definitions from Galois Theory
  • LO9. Determine Galois groups in various examples
  • LO10. Construct rigorous proofs, and manipulate and apply abstract concepts with an emphasis on the clear explanation of such concepts to others