pfield

The p‑field, also known as a prime field, is the smallest subfield of any field whose characteristic is a prime number p. In abstract algebra, a field is a set equipped with two operations, addition and multiplication, satisfying the usual axioms of commutativity, associativity, distributivity, the existence of additive and multiplicative identities, and the existence of additive inverses for all elements and multiplicative inverses for all non‑zero elements. The characteristic of a field is the smallest positive integer n such that adding the multiplicative identity to itself n times yields the additive identity; if no such n exists, the field has characteristic zero. When the characteristic is a prime p, the subfield generated by repeatedly adding the multiplicative identity forms a field with exactly p elements, denoted GF(p) (Galois field of order p) or ℤ/pℤ, the integers modulo p. This p‑field is unique up to isomorphism and serves as the foundational building block for all finite fields of characteristic p, since any such field contains a copy of GF(p) as its prime subfield.

Finite extensions of a p‑field are constructed by adjoining roots of irreducible polynomials over GF(p), resulting in larger fields GF(p^n) of order p^n, where n is a positive integer. These extensions play a central role in number theory, coding theory, and cryptography, particularly in the design of error‑correcting codes and in the implementation of elliptic curve cryptography. In algebraic geometry, the notion of the prime field underlies the definition of varieties over finite fields, influencing the study of rational points and Weil conjectures. The concept of a p‑field thus provides a fundamental link between elementary modular arithmetic and the broader structure of modern algebraic systems.