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

Frank Gerth III

doi:10.4153/CMB-2002-009-8

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

Published online by Cambridge University Press: 20 November 2018

Frank Gerth III

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Frank Gerth III*: Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712-1082, U.S.A., email: gerth@math.utexas.edu

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Abstract

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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.