Z2Z

Z2Z is a common shorthand used to denote the ring of integers modulo 2, formally written as Z/2Z. It is the finite field with two elements, commonly represented as {0, 1}. In this system, arithmetic is performed modulo 2.

As an algebraic structure, Z/2Z is a field. The addition table is 0+0=0, 0+1=1, 1+0=1, 1+1=0, so addition is equivalent to bitwise XOR. The multiplication table is 0×0=0, 0×1=0, 1×0=0, 1×1=1, so multiplication corresponds to ordinary multiplication modulo 2. The characteristic of Z/2Z is 2, meaning 1+1=0. The additive group is cyclic of order 2, and the multiplicative group of nonzero elements is the trivial group {1}.

Z/2Z is the smallest finite field and serves as the prime field of characteristic 2. It forms the base field for constructing binary vector spaces and more complex finite fields, and it underpins many applications in coding theory, cryptography, and digital logic. In binary linear algebra, vectors are elements of F2^n, with operations carried out modulo 2. In computer science, addition over GF(2) corresponds to bitwise XOR, while multiplication is the standard modulo-2 product.

Notationally, Z2Z appears in some literature, but the standard conventions for this object are Z/2Z and GF(2). Regardless of notation, it remains the simplest nontrivial finite field and a foundational element in discrete mathematics and related disciplines.