1. The problem statement, all variables and given/known data Suppose a field F has n elements and [itex]F=(a_1,a_2,...,a_n)[/itex]. Show that the polynomial [itex]w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F[/itex] has no roots in F, where [itex]1_f[/itex] denotes the multiplicative identity in F. 2. Relevant equations 3. The attempt at a solution Strategy: We have this polynomial: [itex]w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F[/itex] When we set it to 0 we see that the (x-a) terms need to equal -1 for w(x) to have a root. But here's my problem. I was thinking of trying to use induction on this proof. But if n = 0, then we have x+1 = 0 and x = i^2 which is in a field, the complex field. That has a root. But ok, let's say n = 1. Then we have x-a+1=0 in which case x = 3, a=4 would be a root. Must I assume that there is more then n=1 elements?