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

Volume 164 / 2014

John C. Miller Acta Arithmetica 164 (2014), 381-397 MSC: Primary 11R29, 11R80; Secondary 11R18, 11Y40. DOI: 10.4064/aa164-4-4


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.


  • John C. MillerDepartment of Mathematics
    Rutgers University
    Hill Center for the Mathematical Sciences
    110 Frelinghuysen Road
    Piscataway, NJ 08854-8019, U.S.A.

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