There are only answers without any reasons in my textbook's questions.
PLEASE HELP ME :(
Find the root of these polynomials by using hints.
Let the $w$ is complex root of $x^2+x+1$
1.First question
$f(x)=x^3 + 6x^2 -2 $ which means $ irr(\alpha,Q)$ (hint : one of the real root of $f(x)$ is $\alpha = \sqrt[3]{4}-\sqrt[3]{2}$ )
Solution in my textbook said that that all the roots are
$\alpha$
$\beta$ =$\sqrt[3]{4}w^2-\sqrt[3]{2}w$,
$\gamma$ = $\sqrt[3]{4}w-\sqrt[3]{2}w^2$'
I'm tried to find the other roots $\beta$ and $\gamma$ But failed. :(
Is there any Either formula or principle finding the $\beta$ and $\gamma$? Please give me some ideas.
2.Second question
$g(x)=x^9 -3x^6+165x^3-1 $ which means $ irr(\alpha,Q)$ (hint : one of the real root of $g(x)$ is $\alpha = \sqrt[3]{3}-\sqrt[3]{2}$ )
Solution in my textbook said that that all the roots form is
$ \sqrt[3]{3}w^n-\sqrt[3]{2}w^m$ [ $m,n=0,1,2$]
All I know the number of roots is 9 thinking the splitting field of $g(x)$ over $Q$ And It might having the similar principle with the case of $f(x)$
BUT I couldn't find exact all the complex roots though $\alpha$ was given.
Why is all the roots of $g(x)$ have a form that $ \sqrt[3]{3}w^n-\sqrt[3]{2}w^m$?
Let $\theta = \sqrt[3]{2}$. Then the conjugates of $\theta$ are $\theta\omega$ and $\theta\omega^2$, where $\omega$ is a primitive cubic root of unity.
Now $\alpha = \theta^2-\theta = h(\theta)$ and so its conjugates are $h(\theta\omega)$ and $h(\theta\omega^2)$. These are the other roots of $f$.
A similar answer works for the second question.