Problem 5

Find all polynomials

P(x)

with integer coefficients such that for all real numbers

s

and

t

, if

P(s)

and

P(t)

are both integers, then

P(st)

is also an integer.

Step 7 of 8: Force the leading coefficient to be 1

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a=b=a_n^{-1/n},\quad P(a)=P(b)=1\in\mathbb Z\ \Longrightarrow\ P(ab)=\frac1{a_n}\in\mathbb Z\ \Longrightarrow\ a_n=1

Detailed analysis

With $P(x)=a_nx^n$ and $a_n>0$ , set $a=b=a_n^{-1/n}$ , so $P(a)=P(b)=a_n\cdot a_n^{-1}=1\in\mathbb Z$ . The hypothesis then forces $P(ab)=P(a^2)=a_n\,a_n^{-2}=1/a_n$ to be an integer. Since $a_n$ is a positive integer, this forces $a_n=1$ .