The 3-adic regulators and wild kernels

Abstract

For any number field, J.-F. Jaulent introduced a new invariant called the group of logarithmic classes in 1994. This invariant is proved to be closely related to the wild kernels of number fields. In this paper, we show how to compute the kernel of the natural homomorphism from the group of logarithmic classes to the group of p-ideal classes by computing the p-adic regulator which is a classical invariant in number theory. As an application, we prove Gangl's conjecture on 9-rank of the tame kernel of imaginary quadratic field Q( √ −9k − 3).

Reflection theorems and the p-Sylow subgroup of K2OF for a n

“…Property (i) was proved by Browkin [2]. An alternative proof for (ii) by using the p-adic regulator and the wild kernel has been obtained by Guo and Qin [7]. (i) if p = 3 is a prime or p = 3, d ≡ 6 (mod 9), then p| K 2 O F implies p|h(Q( √ d(ζ p − ζ −1 p ))) · h(Q(ζ p )), and if moreover d < 0, then p| K 2 O F implies p|h(Q( √ d, ζ p + ζ −1 p )) · h(Q(ζ p ));…”

On the 3-rank of tame kernels of certain pure cubic number fields

“…There are many results describing the structure of K 2 O F , see [2,[17][18][19][20][21]24] and others for the results about the 2-primary part of K 2 O F , when F is a quadratic number field. For an odd prime p, results on the p-primary part of K 2 O F can be found in [7,13,14,16,22,23], etc. In particular, some results on the 3-rank of K 2 O F were given by Browkin [4], Cheng [6], Qin and Zhou [22,23].…”

Tame kernels of quintic cyclic fields

“…Introduction. The structure of the tame kernels of algebraic number fields has been investigated by many authors, e.g., [1], [2], [4], [5], [7], [10], [11], [12] and [15]. In particular, J. Browkin gave some explicit results for cubic cyclic fields with only one ramified prime in [1], and H. Zhou investigated the structure of tame kernels of cubic cyclic fields with two ramified primes in [15].…”