GFq

GF(q), also written GF_q, denotes a finite field with q elements, where q is a power of a prime p. When q is prime, the field is the prime field GF(p), consisting of the integers {0,1,...,p−1} with arithmetic modulo p. For q = p^n with n > 1, GF(q) is the unique finite field of that order up to isomorphism.

GF(q) can be constructed as F_p[x]/(f(x)), where f(x) is an irreducible polynomial over F_p of degree n. In this representation, elements are polynomials of degree less than n with coefficients in F_p, reduced modulo f(x). The additive group of GF(q) is a vector space of dimension n over F_p, while the multiplicative group GF(q)^× is cyclic of order q−1. The Frobenius automorphism x ↦ x^p generates the Galois group of GF(q) over F_p.

Elements of GF(q) can be represented in various forms, including polynomial basis or normal basis. Arithmetic in GF(q) uses addition and multiplication modulo p and the irreducible polynomial, with efficient algorithms for modular reduction, inversion, and exponentiation.

GF(q) fields underpin many practical areas. In coding theory, they enable Reed-Solomon and BCH codes. In cryptography, finite fields underpin elliptic-curve cryptography and discrete-log based systems. They also appear in digital communications, error correction, and data integrity applications. Common examples include GF(p) for prime p and GF(2^m) used in error-correcting codes and cryptographic protocols.