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. Because of the algebraic properties above, many familiar and powerful tools of mathematics work in GF(2) just as well as other fields.

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. This vector space will have a basis, implying that the number of elements of V must be a power of 2 (or infinite). 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. 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 unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.These spaces can also be augmented with a multiplication operation that makes them into a field GF(2 n), 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(2) modulo a irreducible polynomial (as for instance for the field GF(2 8) in the description of the Advanced Encryption Standard cipher). 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). In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1. If your GF2 flight was cancelled or you arrived to Bahrain with a delay of 3 hours or more, you are entitled to 600€ in compensation, according to the EC 261/2004 regulation. 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. Conway realized that F can be identified with the ordinal number ω ω ω {\displaystyle \omega All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property).