My Math Forum Inert in the Maximal Real Subfield of Cyclotomic Field

 Number Theory Number Theory Math Forum

 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

Quote:
 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 Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top