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A FielD, briefly, is an algebraic system of elements, usually called numbers, in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed without leaving the system, the formal associative, commutative, and distributive rules hold, and the equations a=b+x and a=b*y have solutions, without restriction, provided in the latter case a not=0.

Definition: Given a SeT F and two BinaryOperation?s defined on F, called addition and multiplication, denoted "+" and "*" respectively. Then (F,+,*) is called a FielD if:

Note that a FielD is just a RinG where all elements have multiplicative inverses, except 0 which cannot. Examples: Three Common FielDs.

The concept of a FielD is of use, for example, in defining VectorS? and MatriceS?, two structures in LinearAlgebra?, whose components can be elements of an arbitrary FielD.

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Last edited February 16, 2001 12:16 am by JoshuaGrosse (diff)