On imaginary quadratic number fields with 2-class group of rank 4 and infinite 2-class field tower

Elliot Benjamin

Vol. 201 (2001), No. 2, 257–266

DOI: 10.2140/pjm.2001.201.257

Abstract

Let k be an imaginary quadratic number field with C_k,2, the 2-Sylow subgroup of its ideal class group C_k, of rank 4. We show that k has infinite 2-class field tower for particular families of fields k, according to the 4-rank of C_k, the Kronecker symbols of the primes dividing the discriminant Δ_k of k, and the number of negative prime discriminants dividing Δ_k. In particular we show that if the 4-rank of C_k is greater than or equal to 2 and exactly one negative prime discriminant divides Δ_k, then k has infinite 2-class field tower.

Milestones

Received: 9 December 1999

Revised: 19 April 2000

Published: 1 December 2001