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# Field

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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 b <> 0.

Definition: A field is a commutative ring (F,+,*) such that all elements of F except 0 have a multiplicative inverse. This means that the following holds:

Closure of F under + and *
  For all a,b belonging to F both a + b and a * b belong to F.

Both + and * are associative
  For all a,b,c in F, a + (b + c) = (a + b) + c

Both + and * are commutative
  For all a,b belonging to F, a + b = b + a and a * b = b * a.

The operation * is distributive over the operation +
  For all a,b,c, belonging to F, a * (b + c) = (a * b) + (a * c) and (a + b) * c = (a * c) + (b * c).

Existence of an additive identity
  There exists an element 0 in F, such that for all a belonging to F, a + 0 = a and 0 + a= a .

Existence of a multiplicative identity
  There exists an element 1 in F, such that for all a belonging to F, a * 1 = a and 1 * a = a.

Existence of an additive inverse
  For all a belonging to F, there exists an element -a in F, such that a + (-a) = 0 and (-a) + a = 0.

Existence of a multiplicative inverse
  For all a <> 0 belonging to F, there exists an element a-1 in F, such that a * a-1 = 1 and a-1 * a = 1.

#### Three Common Fields.

- The rational numbers Q= { a/b | a, b in Z, b <> 0 } where Z is the set of integers.
- The real numbers R .
- The complex numbers C.

The concept of a field is of use, for example, in defining vectors and matrices, two structures in Linear Algebra, whose components can be elements of an arbitrary field.

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Last edited August 12, 2001 1:43 pm (diff)
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