6,537 research outputs found
On wild ramification in quaternion extensions
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.Comment: 19 pages. This is an extensive revision of the earlier draf
Scaffolds and Generalized Integral Galois Module Structure
Let be a finite, totally ramified -extension of complete local
fields with residue fields of characteristic , and let be a
-algebra acting on . We define the concept of an -scaffold on ,
thereby extending and refining the notion of a Galois scaffold considered in
several previous papers, where was Galois and for
. When a suitable -scaffold exists, we show how to
answer questions generalizing those of classical integral Galois module theory.
We give a necessary and sufficient condition, involving only numerical
parameters, for a given fractional ideal to be free over its associated order
in . We also show how to determine the number of generators required when it
is not free, along with the embedding dimension of the associated order. In the
Galois case, the numerical parameters are the ramification breaks associated
with . We apply these results to biquadratic Galois extensions in
characteristic 2, and to totally and weakly ramified Galois -extensions in
characteristic . We also apply our results to the non-classical situation
where is a finite primitive purely inseparable extension of arbitrary
exponent that is acted on, via a higher derivation (but in many different
ways), by the divided power -Hopf algebra.Comment: Further minor corrections and improvements to exposition. Reference
[BE] updated. To appear in Ann. Inst. Fourier, Grenobl
An Evaluation of Instrumental Variable Strategies for Estimating the Effects of Catholic Schools
Several previous studies have relied on religious affiliation and the proximity to Catholic schools as exogenous sources of variation for identifying the effect of Catholic schooling on a wide variety of outcomes. Using three separate approaches, we examine the validity of these instrumental variables. We find that none of the candidate instruments is a useful source of identification of the Catholic school effect, at least in currently available data sets
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