Exercise 1.4.17* (Integer-valued polynomials again)

Question

Show that if $f(x)$ is an integer-valued polynomial of degree $n$, then $n! f(x)$ is a polynomial with integer coefficients.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

\begin{proof} Let $f(x)$ be an integer-valued polynomial of degree $n$. By Exercise 11, 
$$ f(x) = \sum_{k=0}^n c_k \binom{x}{k} \qquad (c_i \in \R).$$
Since $f(0), \ldots,f(n)$ are integers, Exercise 15 shows that $c_k\in \Z$ for $k= 0,\ldots,n$.
Therefore
$$ n! f(x) = \sum_{k=0}^n c_k \frac{n!}{k!}\,   x(x-1)\cdots (x-k+1).$$
Since $c_k$ and $n!/k!$ are integers for $k=0,\ldots,n$, and $x(x-1)\cdots (x-k+1) \in \Z[x]$, we obtain
$$n!f(x) \in \Z[x].$$
\end{proof}

Exercise 1.4.17* (Integer-valued polynomials again)

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Exercise 1.4.17* (Integer-valued polynomials again)

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