Z5Z

Z5Z denotes the ring of integers modulo 5, commonly written as Z/5Z or GF(5). It consists of five elements: 0, 1, 2, 3, and 4. Arithmetic in Z5Z is carried out with addition and multiplication modulo 5. Because 5 is prime, Z5Z is a field: every nonzero element has a multiplicative inverse. Specifically, 1 inverse is 1, 2 inverse is 3, 3 inverse is 2, and 4 inverse is 4.

The nonzero elements {1, 2, 3, 4} form a multiplicative group of order 4, which is cyclic. For example, 2 is a primitive element, since its successive powers modulo 5 are 2, 4, 3, 1. This ensures that every nonzero element has a unique inverse and that multiplication is well-behaved under modulo 5 reduction.

Applications and context: Z5Z is a standard example of a finite field used in introductory number theory, coding theory, and cryptographic teaching to illustrate modular arithmetic and field properties. It provides a simple setting for exploring operations, inverses, and polynomial arithmetic modulo a prime. In modern notation, the same structure is usually referred to as GF(5) or Z/5Z, with Z5Z appearing less frequently in contemporary texts.

Notes: The term Z5Z can appear in various contexts, but its primary mathematical meaning is the field of integers modulo 5. It embodies the basic principles of modular arithmetic and finite fields of prime order.