Greenberg's conjecture and units in multiple Z_p-extensions

TL;DR

This paper proves Greenberg's pseudo-null conjecture for specific number fields and generalizes Iwasawa's theorem to multiple Z_p-extensions, advancing understanding of units and class groups in p-adic number theory.

Contribution

It establishes the conjecture for fields where p divides the class number and units, and extends Iwasawa's theorem to multiple Z_p-extensions.

Findings

Proved Greenberg's pseudo-null conjecture in new cases.

Generalized Iwasawa's theorem to multiple Z_p-extensions.

Enhanced understanding of units in p-adic extensions.

Abstract

In this paper we prove Greenberg's pseudo-null conjecture for the field of p-th roots of unity in the case that p exactly divides the class number and the index of the global units in the local units. We also generalize to the case of multiple Z_p-extensions a theorem of Iwasawa on the Kummer extension of the cyclotomic Z_p-extension generated by p-power roots of units .