**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).

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

**Existence of additive inverses**

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)

(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.

* 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 **F**_{2} or** Z**_{2}
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 **Z**_{p} = {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]/(X^{2}+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)