Introduces the modern algebraic concepts of a group and a ring. These concepts occur throughout modern mathematics and this paper looks at their properties and some applications.
Modern algebra is studied all over the world - perhaps not surprising in view of its
international beginnings in the late-1700s work of the Swiss mathematician Leonhard
Euler, the French mathematician Joseph Louis Lagrange and the German mathematician
Carl Friedrich Gauss. Their work led to the introduction in the 1800s of the unifying
abstract algebraic concepts of a group and a ring - the first of these pioneered by
the British algebraist Arthur Cayley; the second due to Richard Dedekind, also German.
These two notions of a group (a set with a standard operation, usually called multiplication)
and a ring (a set with two operations, usually called addition and multiplication)
occur throughout modern mathematics in both its pure and applied branches, and even
after more than 100 years since their introduction, most of today's research in modern
algebra involves the study of either groups or rings (or both!).
The learning aims of the paper are principally to develop the notions of a group and ring, to see how these arise in a variety of mathematical settings and to establish their fundamental properties. Since this is a Pure Mathematics paper that will provide the basis for further study in abstract algebra, concepts will be introduced and developed rigorously. We will be doing a lot of proofs!
|Paper title||Modern Algebra|
|Teaching period||First Semester|
|Domestic Tuition Fees (NZD)||$868.95|
|International Tuition Fees (NZD)||$3,656.70|
- MATH 202
- Schedule C
- Arts and Music, Science
- This paper should be of interest to anyone who wishes to see how algebraic properties arising in different branches of pure mathematics can be described using the unifying concepts of a group and a ring. Students who wish to pursue their interests in algebra should take this paper as a foundation to more advanced papers in the theory of groups, Galois Theory, rings, modules and algebras.
- More information link
- View more information about MATH 342
- Teaching staff
- Professor Robert Aldred
- Paper Structure
- Main topics:
- A review of functions; equivalence relations; modular arithmetic
- Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange's theorem; group actions
- Rings; subrings; integral domains; matrix rings; polynomial rings; homomorphism and isomorphism; ideals; quotient rings; The Chinese Remainder theorem
- Teaching Arrangements
- Five lectures per fortnight and one weekly tutorial
- Strongly recommended text: Abstract Algebra, third edition by David S. Dummit and Richard M. Foote.
- Course outline
- View course outline for MATH 342
- Graduate Attributes Emphasised
- Communication, Critical thinking.
View more information about Otago's graduate attributes.
- Learning Outcomes
- Demonstrate the ability to use mathematical reasoning by writing proofs in the context of groups and rings.