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November 16th, 2018, 12:19 PM  #1 
Newbie Joined: Jul 2018 From: morocco Posts: 22 Thanks: 0 Math Focus: algebraic number theory  Parity of the class number of cyclotomic extensions
Hello, please Is there any results in the parity of $\mathbb Q(\zeta_{p^m})$ ( and $Q(\zeta_{2^m})$ ), with $p$ is an odd prime and $m$ a positive integer ??? Thank you 
November 17th, 2018, 10:53 AM  #2 
Newbie Joined: Jul 2018 From: morocco Posts: 22 Thanks: 0 Math Focus: algebraic number theory 
no answers ?!

November 17th, 2018, 11:09 AM  #3 
Senior Member Joined: Aug 2012 Posts: 2,200 Thanks: 645 
Try math.stackexchange.com for questions at that level.

December 9th, 2018, 07:46 PM  #4 
Newbie Joined: Dec 2018 From: Earth Posts: 4 Thanks: 1  Let $h_n$ be the class number of $K_{p^m} = \mathbb Q(\zeta_{p^m})$. If there is a proper subfield of $K_{p^m}$ with even class number, then $h_n$ will be even. With the exceptions of $p=29$ and $p=113$, the first few primes which follow this rule are $p=163, 277, 349, 397, 491, 547, 607, 709, 827, 853, 937, 941$. For each of these primes, there is a proper subfield of $K_{p}$ with even class number. For example, $p=163$, the cubic subfield of $K_{163}$ has class number $4$, and $p=941$, the quintic subfield of $K_{941}$ has class number $16$.
Last edited by vacasquad; December 9th, 2018 at 07:49 PM. 

Tags 
class, cyclotomic, extensions, number, parity 
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