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November 24th, 2018, 09:57 PM   #1
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Math Focus: algebraic number theory
Inert in the Maximal Real Subfield of Cyclotomic Field

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
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December 9th, 2018, 07:25 PM   #2
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Quote:
Originally Posted by Chems View Post
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 $±3 \pmod 8$.
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