Finite field

Definition

A finite field is a field $F$ whose underlying set is finite. The order of $F$ is its number of elements $|F|$.

Remarks

For each prime number $p$, the ring $\mathbb{Z}/p\mathbb{Z}$ is a finite field of order $p$. The order of a finite field is a prime power $p^n$, where $p$ is the characteristic. The field $\mathbb{F}_2[x]/(x^2 + x + 1)$ is a finite field of order $4$. It is not $\mathbb{Z}/4\mathbb{Z}$, which is not a field. Finite fields are used in number theory, algebraic geometry, cryptography and coding theory.

Explanations

L1 human

A finite field is a finite set where you can add, subtract, multiply and divide by nonzero elements, with the usual rules. The simplest ones are the integers modulo a prime.