On wild ramification in quaternion extensions

Details On wild ramification in quaternion extensions On wild ramification in quaternion extensions

2005-11-07

arxiv.org/abs/math/0511176v2

Mathematics

Number Theory

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

Scientific paper

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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