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. 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. Conway realized that F can be identified with the ordinal number ω ω ω {\displaystyle \omega The elements of GF(2) may be identified with the two possible values of a bit and to the boolean values true and false. GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.

This vector space will have a basis, implying that the number of elements of V must be a power of 2 (or infinite). The flight departs London, Heathrow terminal «4» on January 29, 09:30 and arrives Manama/Al Muharraq, Bahrain on January 29, 19:10. 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. 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.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).If the elements of GF(2) are seen as boolean values, then the addition is the same as that of the logical XOR operation. For example, matrix operations, including matrix inversion, can be applied to matrices with elements in GF(2) ( see matrix ring).

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. 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. It follows that GF(2) is fundamental and ubiquitous in computer science and its logical foundations. All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property). 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.Because of the algebraic properties above, many familiar and powerful tools of mathematics work in GF(2) just as well as other fields. 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 bitwise AND is another operation on this vector space, which makes it a Boolean algebra, a structure that underlies all computer science. 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. 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).