# a, b and c are integers. If a polynomial exists so p(a)=b, p(b)=c, and p(c)=a

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#1
show that it cannot have integer coefficients
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5 years ago
#2

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:

This is only possible if , which is clearly impossible if are distinct integers. Contradiction.

So there cannot exist a polynomial that cyclically permute .
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