i may have been confusing them with arithmetic groups, which are a consequence of finite fields.

i just think of finite fields like the old school games where you go across the edge on the right and come back around on the left. the rules about what happens to you in that process though, that's where the devil is. does it wrap around? in what way, let's say it's binary, and we sequence the operation as one pass across the bits, then you can say "yes, this divide threw all the bits to the right one, so we can bring a new bit back from the left now" or if you add two numbers, and it overflows, how do you decide what way those overflowed bits go around the circle? is it a circle, or is it a mirror thing, so if 1010 overflows does it become 0101 on teh other side?