Inert in the Maximal Real Subfield of Cyclotomic Field

Jul 2018
26
0
morocco
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
 
Dec 2018
4
1
Earth
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
The primes that split in $Q_n$ are those congruent to $±1 \pmod {2^{n+1}}$. If $n>3$, then these primes are also congruent to $±1 \pmod 8$, which are also the primes that split in $Q_2$. The inert primes in $Q_2$ are congruent to $±3 \pmod 8$.