On wild ramification in quaternion extensions

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

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 1, page 101-124
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Elder, G. Griffith, and Hooper, Jeffrey J.. "On wild ramification in quaternion extensions." Journal de Théorie des Nombres de Bordeaux 19.1 (2007): 101-124. <http://eudml.org/doc/249946>.

@article{Elder2007,
abstract = {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 $\sqrt\{-1\}$ (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.},
affiliation = {Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 U.S.A.; Department of Mathematics and Statistics Acadia University Wolfville, NS B4P 2R6 Canada},
author = {Elder, G. Griffith, Hooper, Jeffrey J.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {ramification jumps; biquadratic extensions; quaternion extensions},
language = {eng},
number = {1},
pages = {101-124},
publisher = {Université Bordeaux 1},
title = {On wild ramification in quaternion extensions},
url = {http://eudml.org/doc/249946},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Elder, G. Griffith
AU - Hooper, Jeffrey J.
TI - On wild ramification in quaternion extensions
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 1
SP - 101
EP - 124
AB - 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 $\sqrt{-1}$ (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.
LA - eng
KW - ramification jumps; biquadratic extensions; quaternion extensions
UR - http://eudml.org/doc/249946
ER -

References

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  10. C. U. Jensen, N. Yui, Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 155–182. Zbl0691.12011MR977759
  11. O. T. O’Meara, Introduction to quadratic forms. Springer-Verlag, New York, 1971. Zbl0207.05304
  12. H. Reichardt, Über normalkörper mit quaternionengruppe. Math. Z. 41 (1936), 218–221. Zbl62.0169.02MR1545614
  13. J.-P. Serre, Local Fields. Springer-Verlag, New York, 1979. Zbl0423.12016MR554237
  14. E. Witt, Konstruktion von galoisschen körpern der characteristik p zu vorgegebener gruppe der ordnung p f . J. Reine Angew. Math. 174 (1936), 237–245. Zbl0013.19601
  15. B. Wyman, Wildly ramified gamma extensions. Amer. J. Math. 91 (1969), 135–152. Zbl0188.11003MR241386

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