Exersise 7.3.(2) of Escofier Book

Question

Let $a,b,c\in\mathbb{C}$ roots of $f(x)=x^3-3x+1$.Show that $\mathbb{Q}(a)$ is splitting field of $f(x)$ over $\mathbb{Q}$ and represent $b,c$ as linear combination of $\{1,a,a^2\}$.My question is about how can i represent b,c without using Vieta formula?Can you give an alternative solution?

Answer 1

Let $x=2\cos\theta$.

Thus, $$8\cos^3\theta-6\cos\theta+1=0$$ or $$\cos3\theta=-\frac{1}{2},$$ which gives $$\theta\in\{40^{\circ},80^{\circ},160^{\circ}\}$$ and we got following roots:

$-2\cos20^{\circ},$ $2\cos40^{\circ}$ and $2\cos80^{\circ}.$

Let $(a,b,c)=(-2\cos20^{\circ},2\cos40^{\circ},2\cos80^{\circ}).$

Thus, $$b+c=4\cos60^{\circ}\cos20^{\circ}=-a.$$ Now, $$b=2\cos40^{\circ}=2(2\cos^220^{\circ}-1)=a^2-2$$ and $$c=-a-b=-a-a^2+2.$$