Suppose that Irr($\alpha,\mathbb{Q}$) has odd degree and has one unique real root $\alpha$. Calculate Aut($\mathbb{Q}(\alpha^2)/\mathbb{Q}$)

Question

I have been trying to solve this question but I assume there is some relation between the unique real root $\alpha$ and Irr($\alpha^2, \mathbb{Q}$) which I can't seem to figure out. I also don't know wheter $\alpha \in \mathbb{Q}(\alpha^2)$ or not. Any help would be greatly appreciated.