GF(2)
\(\mathrm{GF}(2)\) is the Galois field with only two elements. That is, it contains only the additive and multiplicative identity.
It is, up to isomorphism, the smallest field that exists, since the definition guarantees at least an additive and multiplicative identity, and these elements cannot be equal because otherwise removing the additive identity would not leave a group, as required.
Representation: Logic
One way to represent the finite field of size two is to use \(\mathrm{true}\) and \(\mathrm{false}\) as the two elements, and the binary operations \(\mathrm{XOR}\) and \(\mathrm{AND}\) to be the addition and multiplication of the field respectively.
| XOR | false | true |
| false | false | true |
| true | true | false |
| AND | false | true |
| false | false | false |
| true | false | true |
Representation: Modular Arithmetic
Another way of representing this field is using the set \(\{0, 1\}\) and the usual addition and multiplication operation modulo 2.
| \(+\) | \(0\) | \(1\) |
| \(0\) | \(0\) | \(1\) |
| \(1\) | \(1\) | \(0\) |
| \(\times\) | \(0\) | \(1\) |
| \(0\) | \(0\) | \(0\) |
| \(1\) | \(0\) | \(1\) |
This is equivalent to the other representation given here up to isomorphism, where such an isomorphism is given by: