a, b and c are integers. If a polynomial exists so p(a)=b, p(b)=c, and p(c)=aWatch
Suppose that there does exist a polynomial with integer coefficients that satisfies the property that .
Then is a root of and using the factor theorem yields where is a polynomial with integer coefficients.
Specifically, we have that, , must be an integer.
Similarly, we can see that both numbers and must also be an integer. It is obvious then, that:
, which is clearly impossible if are distinct integers. Contradiction.
So there cannot exist a polynomial that cyclically permute .