On the splitting of the exact sequence relating the wild and tame kernels

TL;DR

This paper investigates the conditions under which the exact sequence linking wild and tame kernels in number fields splits, using Iwasawa theory and relating it to class group invariants, with applications to quadratic fields.

Contribution

It provides new criteria for the splitting of the exact sequence involving wild and tame kernels using Iwasawa theory and class group invariants.

Findings

Splitting conditions are characterized via Iwasawa invariants.

The results are illustrated for quadratic number fields.

The work relates the splitting problem to triviality of certain Galois invariants.

Abstract

Let k be a number field. For an odd prime p and an integer i>1, the i-th \'etale wild kernel is contained in the second cohomology group of o'_k with coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa theory, we give conditions for this inclusion to split. In particular we relate this splitting problem to the triviality of two invariants, namely the asymptotic kernels of the Galois descent and codescent for class groups along the cyclotomic tower of k. We illustrate our results in both split and non-split cases for quadratic number fields.