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Weber’s class number problem in the cyclotomic 2 -extension of . II. (English) Zbl 1223.11133

Let ζ n =exp2πi 2 n , n =(ζ n+2 +ζ n+2 -1 ), h n the class number of n . Let l be an odd prime number, 2 c the exact power of 2 dividing l-1 or l 2 -1 according as l1(mod4) or not, m l =2c-3+[log 2 l]. The authors prove that l does not divide h n h m l for any integer nm l . If l±9(mod16), then l does not divide h n for all n1.

For Part I, see Exp. Math. 18, No. 2, 213–222 (2009; Zbl 1189.11033), Part III, Int. J. Number Theory 7, No. 6, 1627–1635 (2011; Zbl 1226.11119).


MSC:
11R29Class numbers, class groups, discriminants
11R18Cyclotomic extensions