In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1.

F is countable and contains a single copy of each of the finite fields GF(2 n); the copy of GF(2 n) is contained in the copy of GF(2 m) if and only if n divides m. 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. This vector space will have a basis, implying that the number of elements of V must be a power of 2 (or infinite).

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.

The elements of GF(2) may be identified with the two possible values of a bit and to the boolean values true and false.

It follows that GF(2) is fundamental and ubiquitous in computer science and its logical foundations. The bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science. If the elements of GF(2) are seen as boolean values, then the addition is the same as that of the logical XOR operation.

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.

GF(2) can be identified with the field of the integers modulo 2, that is, the quotient ring of the ring of integers Z by the ideal 2 Z of all even numbers: GF(2) = Z/2 Z. The flight departs London, Heathrow terminal «4» on January 29, 09:30 and arrives Manama/Al Muharraq, Bahrain on January 29, 19:10. 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). Conway realized that F can be identified with the ordinal number ω ω ω {\displaystyle \omega

For example, matrix operations, including matrix inversion, can be applied to matrices with elements in GF(2) ( see matrix ring). Gulf Air Flight GF2 connects London, United Kingdom to Bahrain, Bahrain, taking off from London Heathrow Airport LHR and landing at Bahrain International Airport BAH.The multiplication of GF(2) is again the usual multiplication modulo 2 (see the table below), and on boolean variables corresponds to the logical AND operation.