Finite field

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 ).

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

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

2. 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 $0,1,\alpha,\alpha+1$.

3. 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 $8$.