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

<video Finite fields made easy by Randell Heyman (2015)> at https://youtu.be/z9bTzjy4SCg?t=159 shows how for order $9 = 3 \times 3$. Basically, for order $p^n$, we take:
* each element is a polynomial in $GF(p)[x]$, $GF(p)[x]$, the <polynomial over a field>[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 $n$
For a worked out example, see: <GF(4)>.


Finite field of non-prime order