Hello

Let $Q_n$ be the Maximal Real Subfield of of $\mathbb Q(\zeta_{2^{n+1}})$.

for example

$Q_2=\mathbb{\sqrt 2}$ is the maximal real subfield of $\zeta_8$.

How to show that if a rational prime $p$ inert in $Q_2$ then it is inert in $Q_n$.

Thank you