Existence of Finite Fields

A finite field is a finite field .

Theorem (prime power cardinality and existence).

1. If $K$ is a finite field, then its characteristic is a prime $p$, and there exists an integer $n\ge 1$ such that

   $$|K|=p^n.$$

   Concretely, $K$ contains a copy of the prime field $\mathbb{F}_p$, and $K$ is a finite-dimensional vector space over $\mathbb{F}_p$ of dimension $n$, so $|K|=p^n$.

2. Conversely, for every prime $p$ and integer $n\ge 1$ there exists a field of order $p^n$, usually denoted $\mathbb{F}_{p^n}$. One construction is: choose an irreducible polynomial $f(x)\in \mathbb{F}_p[x]$ of degree $n$ and set

   $$\mathbb{F}_{p^n}\ \cong\ \mathbb{F}_p[x]/(f).$$

   This gives a field extension of degree $n$ (see degree of an extension ) generated by a root of $f$, hence a simple extension .

A deeper refinement is that $\mathbb{F}_{p^n}$ is unique up to isomorphism (see existence and uniqueness of finite fields ).

Examples

1. The prime fields $\mathbb{F}_p$.
   For any prime $p$, the quotient ring $\mathbb{Z}/p\mathbb{Z}$ is a field, denoted $\mathbb{F}_p$, and has $p$ elements.

2. A quadratic extension: $\mathbb{F}_4$.
   Over $\mathbb{F}_2$, the polynomial $x^2+x+1$ has no root (so it is irreducible). Thus

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

   whose elements can be written $\{0,1,\alpha,\alpha+1\}$ with $\alpha^2=\alpha+1$.

3. Another quadratic extension: $\mathbb{F}_9$.
   Over $\mathbb{F}_3$, the polynomial $x^2+1$ is irreducible (since $-1\equiv 2$ is not a square in $\mathbb{F}_3$). Hence

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

   and every element has the form $a+b\beta$ with $a,b\in\mathbb{F}_3$ and $\beta^2=-1$.

(As a similar cubic example, one may construct $\mathbb{F}_8$ as $\mathbb{F}_2[x]/(f)$ for any irreducible cubic $f$ over $\mathbb{F}_2$.)