GF(2)

GF(2) (also F2, Z/2Z or Z2) is the Galois field of two elements. It is the smallest finite field.

Definition

The two elements are nearly always called 0 and 1, being the additive and multiplicative identities, respectively.

The field's addition operation is given by the table below, which corresponds to the logical XOR operation.

The field's multiplication operation corresponds to the logical AND operation.

One may also define GF(2) as the quotient ring of the ring of integers Z by the ideal 2Z of all even numbers: GF(2) = Z/2Z.

Properties

Because GF(2) is a field, many of the familiar properties of number systems such as the rational numbers and real numbers are retained:

  • addition has an identity element (0) and an inverse for every element;
  • multiplication has an identity element (1) and an inverse for every element but 0;
  • addition and multiplication are commutative and associative;
  • multiplication is distributive over addition.

    • Properties that are not familiar from the real numbers include:

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