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Relative extensions of number fields and Greenberg’s Generalised Conjecture

Volume 174 / 2016

Sören Kleine Acta Arithmetica 174 (2016), 367-392 MSC: Primary 11R23; Secondary 11R37. DOI: 10.4064/aa8423-5-2016 Published online: 3 August 2016

Abstract

Let $p$ be a fixed prime. In this article, we will prove several results concerning Greenberg’s Generalised Conjecture (GGC). On the one hand, we will prove that whenever a slightly stronger form of (GGC) holds for a number field $K$ (which will be the case in most of the examples), then the conjecture also holds for every finite normal $p$-ramified $p$-extension of $K$. On the other hand, we will directly prove that (GGC) holds for certain number fields containing exactly one prime above $p$. These results are based on the insight that the validity of (GGC) for some number field $K$ can be checked by studying $\mathbb Z_p$- and $\mathbb Z_p^2$-extensions of $K$. We will also provide new examples in which (GGC) holds in a non-trivial way.

Authors

  • Sören KleineInstitut für Theoretische Informatik, Mathematik und Operations Research
    Universität der Bundeswehr München
    Werner-Heisenberg-Weg 39
    D-85577 Neubiberg, Germany
    e-mail

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