Rigidity of valuative trees under henselization

Abstract

. Let ( K, v ) be a valued field and let ( K h , v h ) be the henselization determined by the choice of an extension of v to an algebraic closure of K . Consider an embedding v ( K ∗ ) ֒ → Λ of the value group into a divisible ordered abelian group. Let T ( K, Λ), T ( K h , Λ) be the trees formed by all Λ-valued extensions of v , v h to the polynomial rings K [ x ], K h [ x ], respectively. We show that the natural restriction mapping T ( K h , Λ) → T ( K, Λ) is an isomorphism of posets. As a consequence, the restriction mapping T v h → T v is an isomorphism of posets too, where T v , T v h are the trees whose nodes are the equivalence classes of valuations on K [ x ], K h [ x ] whose restriction to K , K h are equivalent to v , v h , respectively.