# Inert in the Maximal Real Subfield of Cyclotomic Field

#### Chems

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

Let $Q_n$ be the Maximal Real Subfield of of $\mathbb Q(\zeta_{2^{n+1}})$.
$Q_2=\mathbb{\sqrt 2}$ 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 $Q_n$.
The primes that split in $Q_n$ are those congruent to $Â±1 \pmod {2^{n+1}}$. If $n>3$, then these primes are also congruent to $Â±1 \pmod 8$, which are also the primes that split in $Q_2$. The inert primes in $Q_2$ are congruent to $Â±3 \pmod 8$.