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A convenient notation for the elements of of prime order is to use integers, e.g. for 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:
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.
For non-prime prime power orders however, we can find a way, see finite field of non-prime order.
Video 3. Finite fields made easy by Randell Heyman (2015) Source. Good introduction with examples