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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
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 Originally Posted by 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
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 $±3 \pmod 8$. Tags cyclotomic, field, inert, maximal, real, subfield Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post cloudtifa Abstract Algebra 1 December 4th, 2009 06:18 PM mijl43 Linear Algebra 5 July 1st, 2009 10:05 AM brunojo Abstract Algebra 0 June 5th, 2009 06:25 PM bjh5138 Abstract Algebra 1 November 29th, 2007 02:24 PM MathLady Abstract Algebra 1 November 20th, 2006 11:49 PM

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