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 November 16th, 2018, 12:19 PM #1 Newbie   Joined: Jul 2018 From: morocco Posts: 26 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: 26 Thanks: 0 Math Focus: algebraic number theory no answers ?!
 November 17th, 2018, 11:09 AM #3 Senior Member   Joined: Aug 2012 Posts: 2,357 Thanks: 740 Try math.stackexchange.com for questions at that level. Thanks from Chems
December 9th, 2018, 07:46 PM   #4
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 Originally Posted by Chems 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
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.

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