What is a Field?
Definition:
A field is a set  F together with two operations on that set + and *, (F, +, *) that satisfy the following properties:

Closure of F under + and *
For all a,b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F);

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

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 different from 0, such that for all a belonging to F, a * 1 = a and 1 * a = a.

For every a belonging to F, there exists an element b in F, such that a + b = 0 and  b + a = 0. [b is often denoted "-a"]

Existence of multiplicative inverses
For every a not equal to 0 belonging to F, there exists an element b in F, such that a * b = 1 and b * a = 1.  [b is often denoted "a-1" or "1/a"]

The requirement 0 not equal to 1 ensures that the set which only contains a single zero is not a field. Directly from the axioms, one may show that (F, +) and (F - {0}, *) are commutative groups and that therefore (see elementary group theory) the additive inverse -a and the multiplicative inverse a-1 are uniquely determined by a. Furthermore, the multiplicative inverse of a product is equal to the product of the inverses:

(a*b)-1 = a-1 * b-1

provided both a and b are non-zero. Other useful rules include

-a = (-1) * a

and more generally

-(a * b) = (-a) * b = a * (-b)

as well as

a * 0 = 0,

all rules familiar from elementary arithmetic.

Examples of Fields

* The rational numbers Q = { a/b | a, b in Z, b not equal to 0 } where Z is the set of integers.

* The real numbers R .

* The complex numbers C.

* The smallest field has only two elements: 0 and 1. It is sometimes denoted by F2 or Z2 and can be defined by the two tables

+  0  1        *  0  1
0  0  1        0  0  0
1  1  0        1  0  1

It has important uses in computer science, especially in cryptography and coding theory.

* More generally: if q > 1 is a power of a prime number, then there exists (up to isomorphism) exactly one finite field with q elements. No other finite fields exist. For instance, for a prime number p, the set of integers modulo p is a finite field with p elements: this is often written as Zp = {0,1,...,p-1} where the operations are defined by performing the operation in Z, dividing by p and taking the remainder, see modular arithmetic.

* The real numbers contain several interesting fields: the real algebraic numbers, the computable numbers, and the definable numbers.

* The complex numbers contain the field of algebraic numbers, the algebraic closure of Q.

* The rational numbers can be extended to the fields of p-adic numbers for every prime number p.

* Let E and F be two fields with E a subfield of F (i.e., a subset of F containing 0 and 1, closed under the operations + and * of F and with its own operations defined by restriction). Let x be an element of F not in E. Then E(x) is defined to be the smallest subfield of F containing E and x. For instance, Q(i) is the subfield of the complex numbers C consisting of all numbers of the form a+bi where both a and b are rational numbers.

* For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.

* If F is a field, and p(X) is an irreducible polynomial in the polynomial ring F[X], then the quotient field F[X]/(p(X)) is a field with a subfield isomorphic to F. For instance, R[X]/(X2+1) is a field (in fact, it is isomorphic to the field of complex numbers).

* When F is a field, the set F((X)) of formal Laurent series over F is a field.

Simple Theorems

* The set of non-zero elements of a field F is typically denoted by F*,×; it is an abelian group under multiplication. Every finite subgroup of F*, × is cyclic.

* The characteristic of the field (i.e. the smallest positive integer n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1), if exists, is a prime number. Otherwise it is defined as zero.

* The number of elements in finite field is a prime power.

This material was copied and transformed from http://www.wikipedia.org/wiki/Field_(mathematics)