A Generalization of Jakubec's Formula

In 2009 Jakubec gave two determinantal formulas for the relative class number of the pth cyclotomic field, p an odd prime. We generalize one of the formulas to an arbitrary cyclotomic field and also determine the sign of the formula, which he had not given.

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

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.

An extended Stickelberger ideal of the compositum of a bicyclic field and an imaginary quadratic field

We consider certain extension of the Stickelberger ideal of the compositum of a bicyclic field and a quadratic imaginary field, obtained by adding new annihilators to the Stickelberger ideal. We compute the index of this extension, from which we get some divisibility properties for the relative class number of the compositum.