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November 16th, 2018, 01:19 PM   #1
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Parity of the class number of cyclotomic extensions

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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 ???

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November 17th, 2018, 11:53 AM   #2
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November 17th, 2018, 12:09 PM   #3
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December 9th, 2018, 08:46 PM   #4
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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 $16$.
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Last edited by vacasquad; December 9th, 2018 at 08:49 PM.
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