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

John C. Miller¹

¹ Department of Mathematics Rutgers University Hill Center for the Mathematical Sciences 110 Frelinghuysen Road Piscataway, NJ 08854-8019, U.S.A.

Acta Arithmetica, Tome 164 (2014) no. 4, pp. 381-397

Résumé

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.

Détail

DOI : 10.4064/aa164-4-4

Keywords: determination class number totally real fields large discriminant known difficult problem minkowski bound too large useful root discriminant field too large treated odlyzkos discriminant bounds describe technique determining class number fields allowing attack class number problem large class number fields treatable previously known methods application webers class number problem which conjecture real cyclotomic fields power nbsp conductor have class number