GFp

GF(p) denotes a finite field with p elements, where p is a prime number. It is also called the prime field and is denoted F_p or GF(p). The elements can be identified with the integers 0 through p−1, with addition and multiplication performed modulo p. Because p is prime, the set Z/pZ forms a field, meaning every nonzero element has a multiplicative inverse and both addition and multiplication satisfy the field axioms.

Its characteristic is p. The additive group is cyclic of order p, generated by 1. The nonzero elements form a cyclic multiplicative group of order p−1. Therefore, for any a in GF(p) with a ≠ 0, there exists an integer k such that a^k ≡ 1 (mod p).

Arithmetic in GF(p) is performed modulo p. Subtraction is the same as adding the additive inverse modulo p. Division by a nonzero element b means multiplying by the modular inverse of b modulo p, which exists because p is prime.

Examples: GF(5) = {0,1,2,3,4}. 2+4 ≡ 1 (mod 5). 3×4 ≡ 2 (mod 5). The inverse of 2 is 3 since 2×3 ≡ 1 (mod 5).

Applications: GF(p) is used in coding theory, cryptography, and as the base field in many elliptic curves used for public-key cryptography. It also serves as a stepping stone to larger finite fields GF(p^n) constructed as extensions of GF(p).