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GF(2)
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GF(2)
GF(2) (also denoted , Z/2Z or ) is the finite field with two elements.
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 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.
GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.
Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below:
If the elements of GF(2) are seen as Boolean values, then the addition is the same as that of the logical XOR operation. Since each element equals its opposite, subtraction is thus the same operation as addition.
The multiplication of GF(2) is the usual multiplication (see the table below), and on Boolean variables corresponds to the logical AND operation.
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 2Z of all even numbers: GF(2) = Z/2Z.
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GF(2)
GF(2) (also denoted , Z/2Z or ) is the finite field with two elements.
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 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.
GF(2) is the unique field with two elements with its additive and multiplicative identities respectively denoted 0 and 1.
Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below:
If the elements of GF(2) are seen as Boolean values, then the addition is the same as that of the logical XOR operation. Since each element equals its opposite, subtraction is thus the same operation as addition.
The multiplication of GF(2) is the usual multiplication (see the table below), and on Boolean variables corresponds to the logical AND operation.
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 2Z of all even numbers: GF(2) = Z/2Z.