Let a,b,c $\in \mathbb Q$ and $\alpha = \sqrt{a+b\sqrt{c}}$ then all roots of $m_{\alpha,\mathbb Q}(x)$ is in $\mathbb Q(\alpha)$.
I want to show the above question.
But when $\alpha, \sqrt{c} \notin \mathbb Q$, I need to show that $\sqrt{a-b\sqrt{c}} \in \mathbb Q(\alpha)$ because of $m_{a,Q}(x)=x^4-2ax+(a^2-bc^2)$
I want to show the case only with calculation. Help me!