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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.
Study level | Undergraduate |
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Academic unit | Mathematics and Statistics Academic Operations |
Credit points | 6 |
Prerequisites:
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(MATH2922 or MATH2961) or a mark of 65 or greater in (MATH2022 or MATH2061) or 12 credit points 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) |
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Corequisites:
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None |
Prohibitions:
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MATH3062 or MATH3962 |
Assumed knowledge:
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None |
At the completion of this unit, you should be able to:
This section lists the session, attendance modes and locations the unit is available in. There is a unit outline for each of the unit availabilities, which gives you information about the unit including assessment details and a schedule of weekly activities.
The outline is published 2 weeks before the first day of teaching. You can look at previous outlines for a guide to the details of a unit.
Session | MoA ? | Location | Outline ? |
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Semester 1 2024
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Normal day | Camperdown/Darlington, Sydney |
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Session | MoA ? | Location | Outline ? |
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Semester 1 2025
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Normal day | Camperdown/Darlington, Sydney |
Outline unavailable
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Session | MoA ? | Location | Outline ? |
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Semester 1 2020
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Normal day | Camperdown/Darlington, Sydney |
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Semester 1 2021
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Normal day | Camperdown/Darlington, Sydney |
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Semester 1 2021
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Normal day | Remote |
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Semester 1 2022
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Normal day | Camperdown/Darlington, Sydney |
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Semester 1 2022
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Normal day | Remote |
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Semester 1 2023
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Normal day | Camperdown/Darlington, Sydney |
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This refers to the Mode of attendance (MoA) for the unit as it appears when you’re selecting your units in Sydney Student. Find more information about modes of attendance on our website.