normally, is constructed as a quotient ring where f(x) is an irreducible polynomial of degree n in . this lets us think of elements in 3 different ways:
1)polynomial expressions in a root of f(x)
2)a vector space over , of dimension n
3)an extension field of
now isn't n copies of interacting....it's one long loop of length . and although is prime, is not, which means it has divisors. and these divisors "get in the way" when we try to divide:
, which is bad behavior for a field, it means p has no inverse!
geometrically, you can think of it this way: is like n loops of length p, all tied together at 0. is what you get when you untie the loops, tie the end of one to the beginning of the next one, until you have one big long loop. even though we have the same number of elements, it seems unreasonable to expect these will behave "the same".