Overview
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 surprisingly in view of its international beginnings in the late 1700s with 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 Arthur Cayley, the second by Richard Dedekind. 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) are very important in many of the sciences as well as mathematics in both its pure and applied branches. 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 groups and rings, to see how these arise in a variety of mathematical settings, and to establish their fundamental properties. We will also study practical applications of these objects to cryptography and to detecting/correcting errors that occur in transmission of data.
About this paper
| Paper title | Modern Algebra |
|---|---|
| Subject | Mathematics |
| EFTS | 0.15 |
| Points | 18 points |
| Teaching period | Semester 2 (On campus) |
| Domestic Tuition Fees ( NZD ) | $1,103.10 |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Prerequisite
- MATH 202
- Schedule C
- Arts and Music, Science
- Eligibility
This paper will be of interest to anyone who wishes to see how algebraic properties and phenomena arising in different branches of mathematics and science can be described and understood using the concepts of groups and rings, and how these concepts can be applied to contemporary practical problems regarding private and accurate communciation of data across insecure or unreliable channels.
Students who wish to pursue their interests in algebra should take this course as a foundation to more advanced papers in the theory of groups, Galois Theory, rings, modules and algebras.
- Contact
For more information, contact MATH and COMO 200 to 300-level Advisor Associate Professor Jörg Hennig
- Teaching staff
- Paper Structure
Main topics:
- Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange’s theorem, RSA encryption
- Rings; subrings; homomorphism and isomorphism; ideals; quotient rings; integral domains; fields; polynomial rings; factorisation in rings, error-correcting codes
- Teaching Arrangements
Five lectures per fortnight and one weekly tutorial.
- Textbooks
Recommended texts: Abstract Algebra, third edition by David S. Dummit and Richard M. Foote; Contemporary Abstract Algebra, eighth edition by Joseph A. Gallian.
- 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.
Overview
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 surprisingly in view of its international beginnings in the late 1700s with 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 Arthur Cayley, the second by Richard Dedekind.
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) are very important in many of the sciences as well as mathematics in both its pure and applied branches. 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 groups and rings, to see how these arise in a variety of mathematical settings, and to establish their fundamental properties. We will also study practical applications of these objects to cryptography and to detecting/correcting errors that occur in transmission of data.
About this paper
| Paper title | Modern Algebra |
|---|---|
| Subject | Mathematics |
| EFTS | 0.1500 |
| Points | 18 points |
| Teaching period | Semester 2 (On campus) |
| Domestic Tuition Fees | Tuition Fees for 2027 have not yet been set |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Prerequisite
- MATH 202
- Schedule C
- Arts and Music, Science
- Eligibility
This paper will be of interest to anyone who wishes to see how algebraic properties and phenomena arising in different branches of mathematics and science can be described and understood using the concepts of groups and rings, and how these concepts can be applied to contemporary practical problems regarding private and accurate communciation of data across insecure or unreliable channels.
Students who wish to pursue their interests in algebra should take this course as a foundation to more advanced papers in the theory of groups, Galois Theory, rings, modules and algebras.
- Contact
For more information, contact MATH and COMO 200 to 300-level Advisor Associate Professor Jörg Hennig
- Teaching staff
- Paper Structure
Main topics:
- Groups; subgroups; homomorphism and isomorphism; cosets and normal subgroups; quotient groups; Lagrange’s theorem, RSA encryption
- Rings; subrings; homomorphism and isomorphism; ideals; quotient rings; integral domains; fields; polynomial rings; factorisation in rings, error-correcting codes
- Teaching Arrangements
Three lectures and one tutorial per week.
- Textbooks
Recommended Texts:
- Abstract Algebra, third edition by David S. Dummit and Richard M. Foote; Contemporary Abstract Algebra, eighth edition by Joseph A. Gallian.
- Graduate Attributes Emphasised
- Interdisciplinary perspective, Lifelong learning, 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.