A convenient notation for the elements of $GF(n)$ of prime order is to use integers, e.g. for $GF(7)$ we could write:
which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:

$GR(7)={−3,−2,−1,0,1,2,3}$

$GR(7)={0,1,2,3,4,5,6}$

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:
we see that things wrap around perfecly, and 1 is never reached.

$0×2=01×2=22×2=43×2=04×2=25×2=4$

For non-prime prime power orders however, we can find a way, see finite field of non-prime order.