Ciro Santilli @cirosantilli  40

Ciro Santilli really loved his documentary called Can't get you out of my head by Adam Curtis (2021), and then proceeded to basically watch all of this films.

His films Blaise Pascal (1972) and Cartesius (1974) have the greatest scenery and wardrobe ever. The eerie horror movie soundtrack is also very interesting. Ciro was weirdly reminded of Cosmos: A Personal Voyage (1980).

One of the defining properties of algebraic structure with two operations such as ring and field:This property shows how the two operations interact.

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:

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.

A function:

We define this as the functional equation:It is a bit like cauchy's functional equation but with multiplication instead of addition.