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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On Ore's conjecture and its developments


Authors: Ilaria Del Corso and Roberto Dvornicich
Journal: Trans. Amer. Math. Soc. 357 (2005), 3813-3829
MSC (2000): Primary 11R04; Secondary 11R99
Published electronically: April 22, 2005
MathSciNet review: 2146651
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Abstract: The $p$-component of the index of a number field $K$, ${ \rm ind}_p(K)$, depends only on the completions of $K$ at the primes over $p$. More precisely, ${\rm ind}_p(K)$ equals the index of the $\mathbb{Q} _p$-algebra $K\otimes\mathbb{Q} _p$. If $K$ is normal, then $K\otimes\mathbb{Q} _p\cong L^n$ for some $L$ normal over $\mathbb{Q} _p$ and some $n$, and we write $I_p(nL)$ for its index. In this paper we describe an effective procedure to compute $I_p(nL)$ for all $n$ and all normal and tamely ramified extensions $L$ of $\mathbb{Q} _p$, hence to determine ${\rm ind}_p(K)$ for all Galois number fields that are tamely ramified at $p$. Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of $I_p(nL)$.


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Additional Information

Ilaria Del Corso
Affiliation: Dipartimento di Matematica, Università di Pisa, via Buonarroti, 2, 56127 Pisa, Italy
Email: delcorso@dm.unipi.it

Roberto Dvornicich
Affiliation: Dipartimento di Matematica, Università di Pisa, via Buonarroti, 2, 56127 Pisa, Italy
Email: dvornic@dm.unipi.it

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03707-4
PII: S 0002-9947(05)03707-4
Received by editor(s): July 31, 2000
Received by editor(s) in revised form: April 20, 2004
Published electronically: April 22, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.