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Greenberg's conjecture and capitulation in -extensions

✍ Scribed by Andrea Bandini

Publisher: Elsevier Science
Year: 2007
Tongue: English
Weight: 167 KB
Volume: 122
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2006.04.004

No coin nor oath required. For personal study only.

✦ Synopsis

Let p be an odd prime. Let k be an algebraic number field and let k be the compositum of all the Z p -extensions of k, so that Gal( k/k) Z d p for some finite d. We shall consider fields k with Gal(k/Q) (Z/2Z) n . Building on known results for quadratic fields, we shall show that the Galois group of the maximal abelian unramified pro-p-extension of k is pseudo-null for several such k's, thus confirming a conjecture of Greenberg. Moreover we shall see that pseudo-nullity can be achieved quite early, namely in a Z 2 pextension, and explain the consequences of this on the capitulation of ideals in such extensions.

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