Fq

Fq, often written F_q, denotes a finite field with q elements. In most contexts q is a power of a prime, q = p^n, with p prime and n ≥ 1. The field is usually denoted GF(q) or GF(p^n).

Existence and uniqueness: For every prime p and integer n ≥ 1 there exists a finite field of order q = p^n, and up to isomorphism there is a unique field with q elements. This makes Fq a finite field, or Galois field, of characteristic p.

Construction: F_q is an extension of the prime field GF(p). When n > 1 it can be constructed as GF(p)[x]/(f(x)) where f is an irreducible polynomial of degree n over GF(p). Elements are represented by polynomials of degree at most n−1 with coefficients in GF(p). A primitive element α satisfying a chosen irreducible relation provides a convenient generator for the field.

Algebraic structure: The additive group of F_q is a vector space of dimension n over GF(p). The multiplicative group F_q^× is cyclic of order q−1. The Frobenius automorphism x ↦ x^p generates the Galois group of the extension, and for each d dividing n there is a unique subfield GF(p^d).

Applications: Finite fields are central to error detection and correction codes (such as Reed-Solomon and BCH codes), cryptography (including elliptic-curve cryptography), digital signal processing, and various algorithms in computer algebra and communications. A common example is GF(2^8) used in AES and QR codes.

Notes: Because of the finite number of elements, arithmetic in F_q is performed with modular reduction, and there are efficient algorithms for field multiplication, inversion, and exponentiation.