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"]
(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 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.