RINGS AND FIELDS

T he reader is assumed to be familiar with the use of homogeneous and non-homogeneous coordinates in geometry, when the coordinates are real or complex numbers. When geometry is developed with the help of these coordinates, results are obtained by methods which belong to algebra, the differential calculus, and so on. Those results which can be obtained by purely algebraic processes (and they include many which are usually obtained by the methods of the calculus) make up the subject with which we are concerned in this work. The operations of algebra we shall study are those of addition, subtraction, multiplication, division, and the solution of algebraic equations. While ordinary complex numbers are the most familiar elements for which these operations are defined, there are more general sets of elements for which it is possible to define them. By allowing our coordinates to belong to these sets, a more general geometry is obtained. We thus arrive at the definition of a more general space than that considered in elementary geometry, and the study of this space is the purpose of this work. In this and succeeding chapters we consider sets of elements for which some or all of the algebraic operations cited above are defined, and step by step we arrive at a characterisation of the sets of elements from which our coordinates may be chosen. These sets are known in algebra as fields .