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# On Cyclic Fields of Odd Prime Degree $p$ with Infinite Hilbert $p$-Class Field Towers

Published:2002-03-01
Printed: Mar 2002
• Frank Gerth III
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## Abstract

Let $k$ be a cyclic extension of odd prime degree $p$ of the field of rational numbers. If $t$ denotes the number of primes that ramify in $k$, it is known that the Hilbert $p$-class field tower of $k$ is infinite if $t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive proportion of such fields $k$ have infinite Hilbert $p$-class field towers.
 MSC Classifications: 11R29 - Class numbers, class groups, discriminants 11R37 - Class field theory 11R45 - Density theorems

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