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

http://dx.doi.org/10.4153/CMB-2002-009-8
Canad. Math. Bull. 45(2002), 86-88

  Published:2002-03-01
 Printed: Mar 2002

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.