Wir machen den Ansatz

${\displaystyle {}X^{3}+XY^{4}+cY^{6}={\left(X-\alpha Y^{2}\right)}{\left(X-\beta Y^{2}\right)}{\left(X-\gamma Y^{2}\right)}\,.}$

Ausmultiplizieren ergibt

${\displaystyle {}{\left(X-\alpha Y^{2}\right)}{\left(X-\beta Y^{2}\right)}{\left(X-\gamma Y^{2}\right)}=X^{3}-(\alpha +\beta +\gamma )X^{2}Y^{2}+(\alpha \beta +\alpha \gamma +\beta \gamma )XY^{4}-\alpha \beta \gamma Y^{6}\,,}$

was auf die Bedingungen

${\displaystyle {}\alpha +\beta +\gamma =0\,,}$

${\displaystyle {}\alpha \beta +\alpha \gamma +\beta \gamma =1\,}$

und

${\displaystyle {}\alpha \beta \gamma =-c\,}$

führt. Dies bedeutet wiederum (setze ${\displaystyle {}Y=1}$), dass

${\displaystyle {}(X-\alpha )(X-\beta )(X-\gamma )=X^{3}+X+c\,}$

${\displaystyle {}\alpha ,\beta ,\gamma }$

${\displaystyle {}X^{3}+X+c}$