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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*
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.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Cassels, J. and Fröhlich, A., Algebraic Number Theory. Academic Press, London, New York, 1986.Google Scholar
[2] Gerth, F., Densities for ranks of certain parts of p-class groups. Proc. Amer.Math. Soc. 99 (1987), 18.Google Scholar