This tract provides an introduction to four finite geometrical systems and to the theory of projective planes. Of the four geometries, one is based on a nine-element field and the other three can be constructed from the nine-element 'miniquaternion algebra', a simple system which has many though not all the properties of a field. The three systems based on the miniquaternion algebra have widely differing properties; none of them has the homogeneity of structure which characterizes geometry over a field. While these four geometries are the main subject of this book, many of the ideas developed are of much more general significance. The authors have assumed a knowledge of the simpler properties of groups, fields, matrices and transformations (mappings), such as is contained in a first course in abstract algebra. Development of the nine-element field and the miniquaternion system from a prescribed set of properties of the operations of addition and multiplication are covered in an introductory chapter. Exercises of varying difficulty are integrated with the text.
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- Date Published: November 2008
- format: Paperback
- isbn: 9780521090643
- length: 188 pages
- dimensions: 216 x 140 x 11 mm
- weight: 0.25kg
- availability: Available
Table of Contents
Part I. Algebraic Background:
1. Two algebraic systems with nine elements
Part II. Field-Planes:
2. Projective planes
3. Galois planes of orders 3 and 9
Part III. Miniquaternion Planes:
4. The planes Ω and ΩD
5. The plane Ψ.
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