18 research outputs found
Definable valuations on ordered fields
We study the definability of convex valuations on ordered fields, with a
particular focus on the distinguished subclass of henselian valuations. In the
setting of ordered fields, one can consider definability both in the language
of rings and in the richer language of ordered rings
. We analyse and compare definability in both
languages and show the following contrary results: while there are convex
valuations that are definable in the language but
not in the language , any
-definable henselian valuation is already
-definable. To prove the latter, we show that the
value group and the ordered residue field of an ordered henselian valued field
are stably embedded (as an ordered abelian group, respectively as an ordered
field). Moreover, we show that in almost real closed fields any
-definable valuation is henselian.Comment: 17 page
, and division rings of prime characteristic
Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain valued fields of characteristic with Dickson's construction of cyclic algebras, we provide examples of noncommutative division ring of characteristic and show that an division ring of characteristic has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite fields have no Artin-Schreier extension. The result extends to division rings of characteristic , using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern or simple difference fields