On wild ramification in quaternion extensions

2005/11/07 by G. Griffith Elder, Elder, G. Griffith, Jeffrey J. Hooper +1
Mathematics · #11S15 #Algebraic Geometry and Number Theory #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11S15

paper · pdf · doi:10.48550/arxiv.math/0511176

19 pages. This is an extensive revision of the earlier draft

openalex publication_date 2005/11/07 · arxiv created 2006/09/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Quaternion extensions are often the smallest extensions to exhibit special properties. In the setting of the Hasse-Arf Theorem, for instance, quaternion extensions are used to illustrate the fact that upper ramification numbers need not be integers. These extensions play a similar role in Galois module structure. To better understand these examples, we catalog the ramification filtrations that are possible in totally ramified extensions of dyadic number fields. Interestingly, we find that the catalog depends, for sharp lower bounds, upon the refined ramification filtration, which is associated with the biquatratic subfield. Moreover these examples, as counter-examples to the conclusion of Hasse-Arf, occur only when the refined filtration is, in two different ways, extreme.

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