Galois extension and Galois group Let $x$ be a real root of the polynomial $X^3-X+1$, and $y,\overline{y}$ two other roots in $\mathbb{C}$, and $K$ be the cubic field $\mathbb{Q}[x]$. 
Show that $y+\overline{y}=-x$ , $y\overline{y}=-1/x$, and $[(y-x)(\overline{y}-x)(y-\overline{y})]^2=-23$. 
Show that $L=K[\sqrt{-23}]=\mathbb{Q}[x,y,\overline{y}]$ and that field is  a Galois extension of degree $6$ of $\mathbb{Q}$. Determine its Galois group $G$ and subfields of $L$ which are Galois over $\mathbb{Q}$.
I don't now how to solve this. I need help.

Hint: Every extension that contains a complex number has even order, since if it contains $i$ it can be written as $\mathbb Q (\alpha, i) / \mathbb Q (i) / \mathbb Q$. By the tower law, the order of this field over $\mathbb Q$ is $[\mathbb Q (\alpha, i) : \mathbb Q(i)] [\mathbb Q (i) : \mathbb Q]$. Since $[\mathbb Q(i) : \mathbb Q] = 2$, the order of the extension is even. 
