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The cyclic subfield integer index

Bart de Smit (2000)

Journal de théorie des nombres de Bordeaux

In this note we consider the index in the ring of integers of an abelian extension of a number field K of the additive subgroup generated by integers which lie in subfields that are cyclic over K . This index is finite, it only depends on the Galois group and the degree of K , and we give an explicit combinatorial formula for it. When generalizing to more general Dedekind domains, a correction term can be needed if there is an inseparable extension of residue fields. We identify this correction term...

The lifted root number conjecture for fields of prime degree over the rationals: an approach via trees and Euler systems

Cornelius Greither, Radiu Kučera (2002)

Annales de l’institut Fourier

The so-called Lifted Root Number Conjecture is a strengthening of Chinburg’s Ω ( 3 ) - conjecture for Galois extensions K / F of number fields. It is certainly more difficult than the Ω ( 3 ) -localization. Following the lead of Ritter and Weiss, we prove the Lifted Root Number Conjecture for the case that F = ℚ and the degree of K / F is an odd prime, with another small restriction on ramification. The very explicit calculations with cyclotomic units use trees and some classical combinatorics for bookkeeping. An important...

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