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Unit of study_

MATH3962: Rings, Fields and Galois Theory (Adv)

This unit of study investigates the modern mathematical theory that was originally developed for the purpose of studying polynomial equations. The philosophy is that it should be possible to factorize 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 generalizes 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.

Code MATH3962
Academic unit Mathematics and Statistics Academic Operations
Credit points 6
Prerequisites:
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MATH2961 or MATH2922 or a mark of 65 or greater in (MATH2061 or MATH2022)
Corequisites:
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None
Prohibitions:
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MATH3062 or MATH4062

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 applying 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 structured 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 proofs, and manipulate and apply algebraic concepts with an emphasis on the clear explanation of such concepts to others

Unit outlines

Unit outlines will be available 1 week before the first day of teaching for the relevant session.