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Field (mathematics) From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.

All fields are rings, but not conversely. Fields differ from rings most importantly in the requirement that division be possible, but also, in modern definitions, by the requirement that the multiplication operation in a field be commutative. Otherwise the structure is a so-called skew field (better known as a division ring), although historically division rings were called fields and fields were commutative fields.

The prototypical example of a field is Q, the field of rational numbers. Other important examples include the field of real numbers R, the field of complex numbers C and, for any prime number p, the finite field of integers modulo p, denoted Z/pZ, Fp or GF(p). For any field K, the set K(X) of rational functions with coefficients in K is also a field.

The mathematical discipline concerned with the study of fields is called field theory.

Contents [hide] 1 Equivalent definitions 1.1 Definition 1 1.2 Definition 2 1.3 Definition 3 2 History 3 Examples 4 Some first theorems 5 See also 6 References 7 External links

[edit] Equivalent definitions

[edit] Definition 1 A field is a commutative division ring.

[edit] Definition 2 A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. (Note that 0 and 1 here stand for the identity elements for the + and * operations respectively, which may differ from the familiar real numbers 0 and 1.)

[edit] Definition 3

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Ha r ri s o n B Speak! 04:51, 13 September 2007 (UTC)