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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. Tags class, cyclotomic, extensions, number, parity 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 Chems Number Theory 0 October 30th, 2018 10:22 AM acemanhattan Number Theory 1 December 15th, 2013 10:14 AM Rutzer Number Theory 2 April 3rd, 2011 08:01 PM roadnottaken Physics 2 November 3rd, 2007 08:04 PM Infinity Computer Science 12 October 29th, 2007 07:22 PM

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