Existence of Finite Fields

Theorem (Existence).
Let $q=p^n$ be a prime power. There exists a finite field with $q$ elements, denoted $\mathbb{F}_q$.

One construction: choose an irreducible polynomial $g(t)\in \mathbb{F}_p[t]$ of degree $n$ and set

Uniqueness up to isomorphism is treated in existence and uniqueness .

Examples.

1. $q=4$: take $g(t)=t^2+t+1\in\mathbb{F}_2[t]$ (irreducible), so $\mathbb{F}_4\cong \mathbb{F}_2[t]/(t^2+t+1)$.
2. $q=9$: $t^2+1$ is irreducible in $\mathbb{F}_3[t]$, giving $\mathbb{F}_9\cong \mathbb{F}_3[t]/(t^2+1)$.
3. $q=p$: $\mathbb{F}_p\cong \mathbb{Z}/p\mathbb{Z}$ is the prime field of characteristic $p$.

Related. finite-field extensions are cyclic Galois .