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A class number problem in the cyclotomic 3 -extension of . (English) Zbl 1205.11116

Let Ω n =2cos2π 3 n+1 . This is the n-th lyer of the cyclotomic 3 -extension of . Let h n be the class number of Ω n . Let l5 be a prime number and 3 s the exact power of 3 dividing l 2 -1. Put

m l =3s+2+[log 3 (l-1)]+log 3 l-1 2+[log 3 (2s+1+[log 3 (l-1)])],

where [x] denotes the greatest integer not exceeding x. The author proves that if l does not divide h m l , then l does not divide h n for any positive integer n. As a corollary, if l<10000 then l does not divide h n for any positive integer n.

11R23Iwasawa theory
11R29Class numbers, class groups, discriminants
11R18Cyclotomic extensions