Finite Fields: Existence and Uniqueness

Theorem.
Let $q=p^n$ be a prime power. Then:

1. (Existence) There exists a finite field $\mathbb{F}_q$ with exactly $q$ elements (see existence ).
2. (Uniqueness) Any two fields with $q$ elements are isomorphic.

Moreover, $\mathbb{F}_q$ can be realized as $\mathbb{F}_p[t]/(g(t))$ for an irreducible $g$ of degree $n$, and also as the splitting field over $\mathbb{F}_p$ of $x^{q}-x$.

Examples.

1. $\mathbb{F}_4 \cong \mathbb{F}_2[t]/(t^2+t+1)$.
2. $\mathbb{F}_9 \cong \mathbb{F}_3[t]/(t^2+1)$ since $t^2+1$ is irreducible over $\mathbb{F}_3$.
3. $\mathbb{F}_{8} \cong \mathbb{F}_2[t]/(t^3+t+1)$ (any irreducible cubic over $\mathbb{F}_2$ works).

Related. Galois group of finite fields , cyclic multiplicative group .