= Finite field
{tag=Finite algebraic structure}
{title2=$GF(n)$}
{wiki}
A convenient notation for the elements of $GF(n)$ of prime order is to use <integers>, e.g. for $GF(7)$ we could write:
$$
GR(7) = \{-3, -2, -1, 0, 1, 2, 3\}
$$
which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:
$$
GR(7) = \{0, 1, 2, 3, 4, 5, 6\}
$$
For fields of <prime> order, regular <modular arithmetic> works as the field operation.
For non-prime order, we see that <modular arithmetic> does not work because the divisors have no inverse. E.g. at order 6, 2 and 3 have no inverse, e.g. for 2:
$$
0 \times 2 = 0
1 \times 2 = 2
2 \times 2 = 4
3 \times 2 = 0
4 \times 2 = 2
5 \times 2 = 4
$$
we see that things wrap around perfecly, and 1 is never reached.
For non-prime <prime power> orders however, we can find a way, see <finite field of non-prime order>.
\Video[https://www.youtube.com/watch?v=z9bTzjy4SCg]
{title=Finite fields made easy by Randell Heyman (2015)}
{description=Good introduction with examples}