Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and [Formula: see text] the relative class number of Kn. Fixing an integer d ∈ Z with [Formula: see text], we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension [Formula: see text] with Ln ≠ Kn. When d < 0, we have [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that [Formula: see text] is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants [Formula: see text] and [Formula: see text] with d = -1, -3 for all p ≤ 599 and all n with the help of the computer.
On the second factor of the class number of the cyclotomic field
