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Finite field of non-prime order

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

Video 4. "Finite fields made easy by Randell Heyman (2015)" at 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 $G F (p) [x]$ , $G F (p) [x]$ , the polynomial ring over the finite field $G F (p)$ with degree smaller than $n$ . We've just seen how to construct $G F (p)$ for prime $p$ above, so we're good there.
addition works element-wise modulo on $G F (p)$
multiplication is done modulo an irreducible polynomial of order $n$

For a worked out example, see: GF(4).

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