It follows that GF(2) is fundamental and ubiquitous in computer science and its logical foundations.
GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.
Any group (V,+) with the property v + v = 0 for every v in V is necessarily abelian and can be turned into a vector space over GF(2) 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 (or infinite).
In modern computers, data are represented with bit strings of a fixed length, called machine words.
The addition of this vector space is the bitwise operation called XOR (exclusive or).
The bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science.
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(2) modulo a irreducible polynomial (as for instance for the field GF(28) in the description of the Advanced Encryption Standard cipher).
, where the addition and multiplication operations are defined in a natural manner by transfinite induction (these operations are however different from the standard addition and multiplication of ordinal numbers).