Question 25
Let $G$ be the set of polynomials of the form
$$
\begin{aligned} &P(z) = z^n + c_{n-1}z^{n-1} + \cdots \\ &\quad {}+ c_2 z^2 + c_1 z + 50, \end{aligned}
$$
where $c_1,$ $c_2,$ $\ldots,$ $c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a + ib$ with $a$ and $b$ integers. How many polynomials are in $G?$
$$
\begin{aligned} &P(z) = z^n + c_{n-1}z^{n-1} + \cdots \\ &\quad {}+ c_2 z^2 + c_1 z + 50, \end{aligned}
$$
where $c_1,$ $c_2,$ $\ldots,$ $c_{n-1}$ are integers and $P(z)$ has $n$ distinct roots of the form $a + ib$ with $a$ and $b$ integers. How many polynomials are in $G?$