Finite field - Imported Knowlpedia Content

A finite field is a field $F$ with finite cardinality $|F|<\infty$ . Its characteristic is a prime $p$ , so $F$ contains a copy of $\mathbb{F}_p$ , and $F$ is a finite-dimensional field extension of $\mathbb{F}_p$ . Consequently $|F|=p^n$ where $n=[F:\mathbb{F}_p]$ (see degree of an extension).

Remarks

A fundamental classification statement is: for every prime power $q=p^n$ there exists a finite field of order $q$ , and it is unique up to (noncanonical) isomorphism (see existence and uniqueness of finite fields). Moreover, the multiplicative group $F^\times$ is cyclic (see finite-field multiplicative groups are cyclic), and the Frobenius controls the Galois theory of $F/\mathbb{F}_p$ (see finite-field Galois groups are cyclic).

Examples

Prime fields. For a prime $p$ , the field $\mathbb{F}_p=\mathbb{Z}/p\mathbb{Z}$ has $p$ elements.

A field of order $4$ . Let

\mathbb{F}_4 \cong \mathbb{F}_2[t]/(t^2+t+1),

since $t^2+t+1$ is irreducible over $\mathbb{F}_2$ . Writing $\alpha=t\bmod(t^2+t+1)$ , one has $\alpha^2=\alpha+1$ and every element is $0,1,\alpha,\alpha+1$ .

A field of order $9$ . Similarly,

\mathbb{F}_9 \cong \mathbb{F}_3[u]/(u^2+1),

because $u^2+1$ has no root in $\mathbb{F}_3$ . Then $u^2=-1$ and $\mathbb{F}_9^\times$ is cyclic of order $8$ .