Finite field

Definition. A finite field is a field with finitely many elements. Every finite field has cardinality $q=p^n$ for some prime $p$ and integer $n\ge 1$, and (up to isomorphism) there is exactly one field $\mathbb{F}_{q}$ of each such size (see existence and uniqueness of finite fields ).

The multiplicative group $\mathbb{F}_q^\times$ is cyclic (see finite field multiplicative group is cyclic ), and $\mathrm{char}(\mathbb{F}_q)=p$ (see characteristic ).

See also. Frobenius endomorphism , finite field extensions are Galois cyclic .

Examples.

1. $\mathbb{F}_p \cong \mathbb{Z}/p\mathbb{Z}$ for a prime $p$.
2. $\mathbb{F}_4 \cong \mathbb{F}_2[x]/(x^2+x+1)$ since $x^2+x+1$ is irreducible over $\mathbb{F}_2$.
3. $\mathbb{F}_{p^n}$ is the splitting field over $\mathbb{F}_p$ of $x^{p^n}-x$.