Henselian implies large

Abstract

In this note we show that the quotient field of a domain which is Henselian with respect to a nontrivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

Subfields of ample fields. Rational maps and definability

“…Moreover, in recent years ample fields drew attention in several other branches of mathematics -for example in the study of rationally connected varieties and torsors, see [Kol99], [MB01], in the study of Abelian varieties, see [Kob06], [LR08], [FP08], and in the study of definability in fields, see [Koe02], [PP07], [JK08]. We refer the reader to the introduction of [Pop08] for a more extensive survey of ample fields.…”

Hilbertianity of fields of power series

“…] is complete with respect to X , F is ample by [18] and Hilbertian by Theorem 2.4. Thus the result follows by the main theorem of [17].…”

On p-embedding problems in characteristic p

“…The study of the absolute Galois group Gal(K ) of a field K via embedding problems continues to be central in recent papers, e.g. [1,9,10,15,17]. See also the upcoming book [11] and the references therein.…”