GF(2)
is the finite field with two elements.
is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and, as usual.
The elements of may be identified with the two possible values of a bit and to the Boolean values true and false. It follows that is fundamental and ubiquitous in computer science and its logical foundations.
Definition
GF is the unique field with two elements with its additive and multiplicative identities respectively denoted and.Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below:
| + | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 0 |
If the elements of GF are seen as Boolean values, then the addition is the same as that of the logical XOR operation.
Since each element equals its opposite, subtraction is thus the same operation as addition.
The multiplication of GF is the usual multiplication, and on Boolean variables corresponds to the logical AND operation.
| × | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
GF can be identified with the field of the integers modulo, that is, the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers:.
Notations and may be encountered although they can be confused with the notation of -adic integers.
Properties
Because GF is a field, many of the familiar properties of number systems such as the rational numbers and real numbers are retained:- addition has an identity element and an inverse for every element;
- multiplication has an identity element and an inverse for every element but 0;
- addition and multiplication are commutative and associative;
- multiplication is distributive over addition.
- every element x of GF satisfies and therefore ; this means that the characteristic of GF is 2;
- every element x of GF satisfies ; this is an instance of Fermat's little theorem. GF is the only field with this property.
Applications
Any group with the property v + v = 0 for every v in V is necessarily abelian and can be turned into a vector space over GF in a natural fashion, by defining 0v = 0 and 1v = v for all v in V. This vector space will have a basis, implying that the number of elements of V must be a power of 2.
In modern computers, data are represented with bit strings of a fixed length, called machine words. These are endowed with the structure of a vector space over GF. The addition of this vector space is the bitwise operation called XOR. The bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science. These spaces can also be augmented with a multiplication operation that makes them into a field GF, but the multiplication operation cannot be a bitwise operation. When n is itself a power of two, the multiplication operation can be nim-multiplication; alternatively, for any n, one can use multiplication of polynomials over GF modulo a irreducible polynomial.
Vector spaces and polynomial rings over GF are widely used in coding theory, and in particular in error correcting codes and modern cryptography. For example, many common error correcting codes are linear codes over GF, or polynomial codes.
Algebraic closure
Like any field, GF has an algebraic closure. This is a field F which contains GF as a subfield, which is algebraic over GF, and which is algebraically closed. The field F is uniquely determined by these properties, up to a field automorphism.F is countable and contains a single copy of each of the finite fields GF; the copy of GF is contained in the copy of GF if and only if n divides m. The field F is countable and is the union of all these finite fields.
Conway proved that F is isomorphic to the field of ordinals below, with the addition and multiplication operations defined by nimber addition and nimber multiplication respectively. The addition in this field is simple to perform, and Lenstra has shown that the multiplication can also be performed efficiently.