As per classification of finite fields those must be of prime power order.

Video 3. "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order $9=3×3$. Basically, for order $p_{n}$, we take:For a worked out example, see: GF(4).

- each element is a polynomial in $GF(p)[x]$, $GF(p)[x]$, the polynomial ring over the finite field $GF(p)$ with degree smaller than $n$. We've just seen how to construct $GF(p)$ for prime $p$ above, so we're good there.
- addition works element-wise modulo on $GF(p)$
- multiplication is done modulo an irreducible polynomial of order $n$