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November 24th, 2018, 09:57 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 26 Thanks: 0 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 
December 9th, 2018, 07:25 PM  #2 
Newbie Joined: Dec 2018 From: Earth Posts: 4 Thanks: 1  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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cyclotomic, field, inert, maximal, real, subfield 
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