A generalized fundamental principle

Abstract

Let ν be a rank 1 henselian valuation of a field K having unique extension ῡ to an algebraic closure K̅ of K. For any subextension L/K of K̅/K, let G (L), Res (L) denote respectively the value group and the residue field of the valuation obtained by restricting ῡ to L. If a ∑ K̅/K define δ_K(a)=sup{ν̅(a-c)|c∑K̅,[K(C):K]<[K(a):K]} ω_K(a)=max{ν̅(a-a¹)|a¹≠a runs over K-conjugates of a}.