<s>[INST] find a polynomial whose roots are inverse of squares of roots of $x^3+px+q$ Question is :
Given a polynomial $f(x)=x^3+px+q\in \mathbb{Q}[x]$ find a polynomial whose roots are inverse of sqares of roots of $f(x)$
Supposing $a,b,c$ as roots of $f(x)$ we have :


*

*$a+b+c=0$

*$ab+bc+ca=p$

*$abc=-q$


Now i need to know what 


*

*$\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}$

*$\dfrac{1}{a^2}\cdot\dfrac{1}{b^2}+\dfrac{1}{b^2}\cdot\dfrac{1}{c^2}+\dfrac{1}{c^2}\cdot\dfrac{1}{a^2}$

*$\dfrac{1}{(abc)^2}$


All i have to do is use $(a+b+c)^2$ formula and others and conclude what those sums,products are.. I am fairly comfortable with that...
But then, this question was from a Galois theory course.. So, i some how guess there is a better way to do this...
Can some one suggest something..
 [/INST] *

*$x^3+px-q$ has roots $-a,-b,-c$.

*$(x^3+px+q)(x^3+px-q)$ has roots $a,b,c,-a,-b,-c$.  

*$x^3+2px^2+p^2x-q^2$ has roots $a^2,b^2,c^2$.

*$1+2px+p^2x^2-q^2x^3$ has roots $1/a^2,1/b^2,1/c^2$.

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