
Number Theory Number Theory Math Forum 
 LinkBack  Thread Tools  Display Modes 
November 24th, 2018, 10:57 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 21 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, 08:25 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 $±3 \pmod 8$.


Tags 
cyclotomic, field, inert, maximal, real, subfield 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Extension field and subfield  cloudtifa  Abstract Algebra  1  December 4th, 2009 07:18 PM 
Linear independence of vectors in the Real field  mijl43  Linear Algebra  5  July 1st, 2009 11:05 AM 
Every quadratic field is contained in a cyclotomic field  brunojo  Abstract Algebra  0  June 5th, 2009 07:25 PM 
subfield in a ring  bjh5138  Abstract Algebra  1  November 29th, 2007 03:24 PM 
Number of Subfield  MathLady  Abstract Algebra  1  November 21st, 2006 12:49 AM 