Hi, Victor. I was thinking about David Wilson's mapping problem and decided that it would be interesting to reformulate it somewhat less randomly as follows: Given the ring Z_n (Z/nZ) and any polynomial P(x) in Z[x], find the cycles from iterating P on Z_n. OK, that would be interesting. But what is the (or a) set of polynomials that determine *distinct* self-mappings of Z_n ??? After all there are only n^n self-mappings in all. Clearly polynomials P(x) and Q(x) are equal on Z_n when their difference vanishes everywhere. Aha, so look at that ideal — call it I_n — of all P(x) vanishing on every element of Z_n. Question: What is I_n ??? P(x) is a P.I.D. so I_n must be generated by one polynomial, so a unique monic one. What is it ??? Then of course the distinct polynomials correspond to the quotient Z[x] / I_n. What does that look like? *Then* we could start finding the cycle structure corresponding to each element of this quotient. ----- And I suppose one could ask the same question of any finite field F(p^n). Or would that be trivial? —Dan