= GF(4)
{c}
{wiki}
<Ciro Santilli> tried to https://en.wikipedia.org/w/index.php?title=Finite_field&type=revision&diff=1044934168&oldid=1044905041[add this example to Wikipedia], but it was reverted, so here we are, see also: <Deletionism on Wikipedia>{full}.
This is a good first example of a field of a <finite field of non-prime order>, this one is a <prime power> order instead.
$4 = 2^2$, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over <GF(2)>:
* 0X + 0
* 0X + 1
* 1X + 0
* 1X + 1
Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of <prime> order such as <GF(2)>, because we are dealing with <prime powers> only. And we have already defined fields of prime order easily previously with <modular arithmetic>.
Over GF(2), there is only one <irreducible polynomial> of degree 2:
$$
X^2+X+1
$$
Addition is defined element-wise with <modular arithmetic> modulo 2 as defined over GF(2), e.g.:
$$
(1X + 0) + (1X + 1) = (1 + 1)X + (0 + 1) = 0X + 1
$$
Multiplication is done modulo $X^2+X+1$, which ensures that the result is also of degree 1.
For example first we do a regular multiplication:
$$
(1X + 0) \times (1X + 1) = (1 \times 1)X^2 + (1 \times 1)X + (0 \times 1)X + (0 \times 1) = 1X^2 + 1X + 0
$$
Without modulo, that would not be one of the elements of the field anymore due to the $1X^2$!
So we take the modulo, we note that:
$$
1X^2 + 1X + 0 = 1(X^2+X+1) + (0X + 1)
$$
and by the definition of modulo:
$$
(1X^2 + 1X + 0) \mod (X^2+X+1) = (0X + 1)
$$
which is the final result of the multiplication.
TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.