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: 2005-09-11
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.576
@article{JTNB_2007__19_1_101_0,
     author = {G. Griffith Elder and Jeffrey J. Hooper},
     title = {On wild ramification in quaternion extensions},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {101-124},
     doi = {10.5802/jtnb.576},
     zbl = {1123.11037},
     mrnumber = {2332056},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_101_0/}
}
Elder, G. Griffith; Hooper, Jeffrey J. 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/item/JTNB_2007__19_1_101_0/

[1] N. P. Byott, G. G. Elder, Biquadratic extensions with one break. Canadian Math. Bull. 45 (2002) no. 2, 168–179. | MR 1904081 | Zbl 1033.11054

[2] N. P. Byott, G. G. Elder, New ramification breaks and additive Galois structure. J. Théor. Nombres Bordeaux 17 (2005) no. 1, 87–07, Les XXIIIémes Journées Arithmetiques (Graz, 2003). | Numdam | MR 2152213 | Zbl 02205434

[3] G. G. Elder, Galois module structure in wildly ramified cyclic extensions of degree p 2 . Ann. Inst. Fourier (Grenoble) 45 (1995) no. 3, 625–647. errata ibid. 48 (1998) no. 2, 609–610. | Numdam | Zbl 0820.11070

[4] G. G. Elder, Galois module structure of ambiguous ideals in biquadratic extensions. Canad. J. Math. 50 (1998), no. 5, 1007–1047. | MR 1650942 | Zbl 1015.11056

[5] G. G. Elder, The Galois structure of ambiguous ideals in cyclic extensions of degree 8. Noncommutative algebra and geometry, 63–89, Lect. Notes Pure Appl. Math., 243, Chapman & Hall/CRC, Boca Raton, FL, 2006. | MR 2189987 | Zbl 1102.11062

[6] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions. Trans. of Math. Monographs, 121, 2nd Ed., American Mathematical Society, Providence, RI, 2002. | MR 1915966 | Zbl 01793794

[7] J.-M.  Fontaine, Groupes de ramification et représentations d’Artin. Ann. Sci. École Norm. Sup. (4) 4 (1971), 337–392. | Numdam | Zbl 0232.12006

[8] A. Fröhlich, Galois module structure of algebraic integers. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 1, Springer-Verlag, Berlin, 1983. | MR 717033 | Zbl 0501.12012

[9] H. Hasse, Number theory. Classics in Mathematics, Springer-Verlag, Berlin, 2002, Reprint of 1980 English ed. Edited with a preface by H. G. Zimmer. | MR 1885791 | Zbl 0991.11001

[10] C. U. Jensen, N. Yui, Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 155–182. | MR 977759 | Zbl 0691.12011

[11] O. T. O’Meara, Introduction to quadratic forms. Springer-Verlag, New York, 1971. | Zbl 0207.05304

[12] H. Reichardt, Über normalkörper mit quaternionengruppe. Math. Z. 41 (1936), 218–221. | MR 1545614

[13] J.-P. Serre, Local Fields. Springer-Verlag, New York, 1979. | MR 554237 | Zbl 0423.12016

[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.

[15] B. Wyman, Wildly ramified gamma extensions. Amer. J. Math. 91 (1969), 135–152. | MR 241386 | Zbl 0188.11003