Solution
= Solution
Set $Y=X+1$. The <polynomial> becomes $Y^4+Y+1$, and translating all roots leaves their pairwise differences unchanged. Part (b) gives
$$
\boxed{\Delta_f=256-27=229.}
$$
It is nonzero and lies strictly between $15^2=225$ and $16^2=256$. An <integer> which is a rational square is an <integer> square, so this is not a rational square. The roots are distinct because the <discriminant> is nonzero. By (c), $\boxed{G_f\not\subseteq A_4}$.