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# Finite field ()

A convenient notation for the elements of of prime order is to use integers, e.g. for we could write: $$GR(7)={−3,−2,−1,0,1,2,3} (2)$$ which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used: $$GR(7)={0,1,2,3,4,5,6} (3)$$
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: $$0×2=01×2=22×2=43×2=04×2=25×2=4 (4)$$ 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.