  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.
