FP

Fp denotes the prime field with p elements, where p is a prime number. It is the finite field of order p in which arithmetic is performed modulo p. The elements can be represented by the integers {0, 1, ..., p−1}, and addition and multiplication are taken modulo p. This structure satisfies the field axioms: every nonzero element has a multiplicative inverse, and the operations are associative, commutative, and distributive.

Fp is also described as Z/pZ, the quotient ring of the integers by the ideal generated by p. It is isomorphic to the prime subfield of any field of characteristic p and is denoted GF(p) in many contexts. The characteristic of Fp is p, meaning that adding the multiplicative identity to itself p times yields zero.

Common examples include F2 = {0, 1} with arithmetic mod 2 and F5 = {0, 1, 2, 3, 4} with arithmetic mod 5. The prime field Fp serves as a base field in various mathematical and computational settings. Arithmetic in Fp underlies many algorithms in number theory, such as modular exponentiation and primality testing, and forms the foundation for cryptographic protocols and error-correcting codes that rely on arithmetic in finite fields.

Generalizations extend to finite fields of order p^n for n > 1, denoted GF(p^n). These larger fields are built as extension fields over Fp using irreducible polynomials of degree n and are widely used in more advanced coding theory, cryptography, and algebraic geometry.