\documentclass{article}
\usepackage{amsmath}
\usepackage{pslatex}
 \renewcommand{\labelenumi}{(\alph{enumi})}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Q}{\mathbf{Q}}
\begin{document}

\noindent 
Which polynomials $f(x) \in \Q[x]$ have the property that $f(n)$ is an
integer for all integers $n$?

\begin{enumerate}
\item For each integer $j \geq 0$, define $p_{j}(x) \in \Q [x]$ by
\[
p_{j}(x) = \frac{x (x-1) (x-2) \dots (x-j+1)}{j!}.
\]
(When $j=0$, $p_{0}(x)=1$.)  Show that any polynomial in $\Q [x]$ can
be written as a rational linear combination of the polynomials
$p_{j}(x)$.
\item Let $I$ be this set of polynomials:
\[
I = \{f(x) \in \Q [x] \,:\, f(n) \in \Z \ \text{for all} \ n \in \Z
\}.
\]
Show that $f(x) \in I$ if and only if $f(x)$ is an integer linear
combination of the polynomials $p_{j}(x)$.  [Hint: if $f(x) \in I$,
evaluate $f(x)$ at the integers $0$, $1$, \dots $k$.]
\end{enumerate}

\noindent 
(This is from D'Angelo and West, \emph{Mathematical Thinking}, 2nd
edition, problem 5.65.)

\end{document}


% From D'Angelo and West, \emph{Mathematical Thinking}, second edition,
% problem 5.65.

% Question: which polynomials $f(x) \in \Q[x]$ have the property that
% $f(n)$ is an integer for all integers $n$?

% Let $I$ be the set of such polynomials.

% \begin{enumerate}
% \item Show that $I$ is closed under sums and under scalar
% multiplication by integers.
% \item For any integer $j \geq 0$, define $p_{j}(x)$ to be the
% polynomial ``$\binom{x}{j}$''; that is,
% \[
% p_{j}(x) = \frac{x (x-1) (x-2) \dots (x-j+1)}{j!}.
% \]
% Show that $p_{j}(x)$ is in $I$ for each $j$.
% \item Show that any polynomial in $\Q [x]$ can be written as a
% rational linear combination of the polynomials $p_{j}(x)$.
% \item Show that $f(x) \in I$ if and only if $f(x)$ is an integer
% linear combination of the polynomials $p_{j}(x)$.  [Hint: if $f(x) \in
% I$, evaluate $f(x)$ at the integers $0$, $1$, \dots $k$.]
% \end{enumerate}
