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. 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).

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

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. If the elements of GF(2) are seen as boolean values, then the addition is the same as that of the logical XOR operation.

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.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. 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). The flight departs London, Heathrow terminal «4» on January 29, 09:30 and arrives Manama/Al Muharraq, Bahrain on January 29, 19:10.

All larger fields contain elements other than 0 and 1, and those elements cannot satisfy this property). 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. In the latter case, x must have a multiplicative inverse, in which case dividing both sides by x gives x = 1.The elements of GF(2) may be identified with the two possible values of a bit and to the boolean values true and false.