Rings, Fields and Galois Theory (Adv) (MATH3962)

UNIT OF STUDY

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.

Our courses that offer this unit of study

Further unit of study information

Classes

Three 1 hour lectures and one 1 hour tutorial per week.

Assessment

One 2 hour exam, homework assignments (100%)

Faculty/department permission required?

No

Unit of study rules

Prerequisites and assumed knowledge

12 credit points of Intermediate Mathematics with average grade of at least credit

MATH2961

Prohibitions

MATH3902, MATH3062, MATH3002

Study this unit outside a degree

Non-award/non-degree study

If you wish to undertake one or more units of study (subjects) for your own interest but not towards a degree, you may enrol in single units as a non-award student.

Cross-institutional study

If you are from another Australian tertiary institution you may be permitted to underake cross-institutional study in one or more units of study at the University of Sydney.