Class numbers of totally real fields and applications to the Weber class number problem

Abstract

The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. We describe a new technique for determining the class number of such fields, allowing us to attack the class number problem for a large class of number fields not treatable by previously known methods. We give an application to Weber's class number problem, which is the conjecture that all real cyclotomic fields of power of 2 conductor have class number 1.

Recovering Short Generators of Principal Ideals in Cyclotomic Rings

“…For our purposes, we need h + (m) not to be very big. For all power-of-two m up to m = 256, and also for m = 512 under GRH, it is known that h + (m) = 1 (see [Mil14]). Whether h + (m) = 1 for all power-of-two m is known as Weber's class number problem, and is presented in the literature as a reasonable conjecture.…”

Class numbers in cyclotomic Zp-extensions

“…However, the root discriminant of B 13,1 is too large for the class number to be calculated unconditionally using van der Linden's methods, and the field B 17,1 has root discriminant too large to be treated by those methods, even under the assumption of GRH. In the author's recent paper [15], an analytic class number upper bound was developed that employed counting prime ideals of the Hilbert class field. Using this new bound, it is possible to unconditionally determine the class number of number fields of discriminant too large to have been treated by previously known methods.…”

“…The strategy is to find sufficiently many generators of totally split principal prime ideals, so that we have sufficiently many primes to include in our set S. Then we can apply the lemma and establish an upper bound for the class number. The reader is invited to consult [15] for more details regarding this method.…”

“…By using the analytic upper bound for the class number developed in [15], we will prove unconditionally that several more fields have class number 1. Furthermore, using SageMathCloud™ [21], which calls underlying routines in the Pari library [22], we calculated that the fields B 29,1 and B 31,1 each have class number 1, under the assumption of the generalized Riemann hypothesis.…”

Height and Weber’s Class Number Problem

“…Later, several authors showed h n = 1 for (p, n) = (2, 4), (2,5), (3,1), (3,2), (3,3), (5,1) and (7, 1) (see [1], [3], [18] and [19]). And recently, J. C. Miller obtained striking results determining h n = 1 for (p, n) = (2, 6), (5,2), (11,1), (13,1), (17,1) and (19,1) (see [20] and [21]). However, calculating one class number by one gives information on the class numbers for only finitely many layers.…”