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In this article the
-essential dimension of generic symbols over fields of characteristic
is studied. In particular, the
-essential dimension of the length
-symbol of degree
is bounded below by
when the base field is algebraically closed of characteristic
. The proof uses new techniques for working with residues in Milne–Kato
-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on
-symbol algebras (i.e, degree 2 symbols) result from this work. The generic
-symbol algebra of length
is shown to have
-essential dimension equal to
-torsion Brauer class. The second is a lower bound of
-essential dimension of the functor
. Roughly speaking this says that you will need at least
independent parameters to be able to specify any given algebra of degree
over a field of characteristic
and improves on the previously established lower bound of 3.
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