TY - JOUR
TI - Class numbers of totally real number fields
DO - https://doi.org/doi:10.7282/T3474CQ7
PY - 2015
AB - 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. This thesis describes a new approach. By finding nontrivial lower bounds for sums over prime ideals of the Hilbert class field, we establish upper bounds for class numbers of fields of larger discriminant. This analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to determine the class numbers of many number fields that have previously been untreatable by any known method. For example, we consider the cyclotomic fields and their real subfields. Surprisingly, the class numbers of cyclotomic fields have only been determined for fields of small conductor, e.g. for prime conductors up to 67, due to the problem of finding the class number of its maximal real subfield, a problem first considered by Kummer. Our results have significantly improved the situation. We also study the cyclotomic Z_p-extensions of the rationals. Based on the heuristics of Cohen and Lenstra, and refined by new results on class numbers of particular fields, we provide evidence for the following conjecture first suggested by Coates: For all primes p, every number field in a cyclotomic Z_p-extension of Q has class number 1.
KW - Mathematics
KW - Cyclotomy
KW - Algebraic fields
LA - eng
ER -