On wild ramification in quaternion extensions
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 101-124.

This paper provides a complete catalog of the break numbers that occur in the ramification filtration of fully and thus wildly ramified quaternion extensions of dyadic number fields which contain -1 (along with some partial results for the more general case). This catalog depends upon the refined ramification filtration, which as defined in [2] is associated with the biquadratic subfield. Moreover we find that quaternion counter-examples to the conclusion of the Hasse-Arf Theorem are extremely rare and can occur only when the refined ramification filtration is, in two different ways, extreme.

Cet article fournit un catalogue complet des nombres de ramification qui se produisent dans la filtration de ramification des extensions totalement ramifiées des corps de nombres dyadiques qui contiennent -1, et dont le groupe Galois est isomorphe au groupe des quaternions (avec quelques résultats partiels dans le cas plus général). Ce catalogue dépend d’un rafinement de la filtration de ramification. Cette filtration était definie dans [2] comme associée au sous-corps biquadratique. En outre, nous montrons que les contre-exemples de type quaternion aux conclusions du théorème de Hasse-Arf sont extrêmement rares et ne peuvent se produire que seulement dans le cas où la filtration raffinée de ramification est extrême dans deux directions distinctes.

Received:
Published online:
DOI: 10.5802/jtnb.576
G. Griffith Elder 1; Jeffrey J. Hooper 2

1 Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 U.S.A.
2 Department of Mathematics and Statistics Acadia University Wolfville, NS B4P 2R6 Canada
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G. Griffith Elder; Jeffrey J. Hooper. On wild ramification in quaternion extensions. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 101-124. doi : 10.5802/jtnb.576. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.576/

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