Panasonic Lumix GF2 Digital Camera with 14mm & 14-42mm Lenses - Black

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GF(2) (also denoted F 2 {\displaystyle \mathbb {F} _{2}} , Z/2 Z or Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ) is the finite field with two elements [1] (GF is the initialism of Galois field, another name for finite fields).

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 0 v = 0 and 1 v = v for all v in V.GF(2) is the field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively 0 and 1, as usual. The bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science.

In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1. 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(2 8) in the description of the Advanced Encryption Standard cipher).The elements of GF(2) may be identified with the two possible values of a bit and to the boolean values true and false. This vector space will have a basis, implying that the number of elements of V must be a power of 2 (or infinite). Conway realized that F can be identified with the ordinal number ω ω ω {\displaystyle \omega Notations Z 2 and Z 2 {\displaystyle \mathbb {Z} _{2}} may be encountered although they can be confused with the notation of 2-adic integers.