42 research outputs found
Uniform definability of henselian valuation rings in the Macintyre language
We discuss definability of henselian valuation rings in the Macintyre
language , the language of rings expanded by n-th power
predicates. In particular, we show that henselian valuation rings with finite
or Hilbertian residue field are uniformly --definable in
, and henselian valuation rings with value group
are uniformly --definable in the ring
language, but not uniformly --definable in
. We apply these results to local fields
and , as well as to higher dimensional local fields
Random Galois extensions of Hilbertian fields
Let L be a Galois extension of a countable Hilbertian field K. Although L
need not be Hilbertian, we prove that an abundance of large Galois
subextensions of L/K are
Hilbertian fields and Galois representations
We prove a new Hilbertianity criterion for fields in towers whose steps are
Galois with Galois group either abelian or a product of finite simple groups.
We then apply this criterion to fields arising from Galois representations. In
particular we settle a conjecture of Jarden on abelian varieties.Comment: 18 pages, accepted for publication in Journal f\"ur die reine und
angewandte Mathemati