<s>[INST] What are the integer coeffcients of a cubic polynomial having two particular properties? Let $f(x) = x^3 + a x^2 + b x + c$ and $g(x) = x^3 + b x^2 + c x  + a\,$ where $a, b, c$ are integers and $c\neq 0\,$. Suppose that the following conditions hold:


*

*$f(1)=0$ 

*The roots of $g(x)$ are squares of the roots of $f(x)$.


I'd like to find $a, b$ and $c$.
I tried solving equations made using condition 1. and relation between the roots, but couldn't solve. 
The equation which I got in $c$ is $c^4 + c^2 +3 c-1=0$ (edit: eqn is wrong). 
Also I was able to express $a$ and $b$ in terms of $c$. But the equation isn't solvable by hand. 
 [/INST] Let $u$, $v$ and $w$ be the roots of $f$, so that $u^2$, $v^2$ and $w^2$ are the roots of $g$. Then comparing the coefficients of
$$(x-u)(x-v)(x-w)=f(x)=x^3+ax^2+bx+c,$$
$$(x-u^2)(x-v^2)(x-w^2)=g(x)=x^3+bx^2+cx+a,$$
yields the equations
\begin{eqnarray*}
a&=&-u-v-w&=&-u^2v^2w^2,\\
b&=&uv+uw+vw&=&-u^2-v^2-w^2,\\
c&=&-uvw&=&u^2v^2+u^2w^2+v^2w^2.
\end{eqnarray*}
This immediately shows that $a=-c^2$, and the identities
\begin{eqnarray*}
u^2+v^2+w^2&=&(u+v+w)^2-2(uv+uw+vw),\\
u^2v^2+u^2w^2+v^2w^2&=&uvw(u+v+w)-(uv+uw+vw)^2,
\end{eqnarray*}
show that $-b=a^2-2b$ and $c=ac-b^2$, respectively, hence $b=a^2=c^4$ and so
$$f(x)=x^3-c^2x^2+c^4x+c,$$
for some $c$. Then $f(1)=1$ implies that
$$c^4-c^2+c+1=0,$$
which has the clear root $c=-1$.  Then $a=-1$ and $b=1$.
</s>