On Saturated Distinguished Chains over a Local Field

Abstract

For a separable element α over a local field K, we consider a sequence (α, α1, …, αn) of elements such that αi is of minimal degree over K with v(αi−1−αi)=sup{v(αi−1−β)∣[K(β):K]<[K(αi−1):K]} and that αn belongs to K, where α0=α and v is the unique valuation on the algebraic closure of K with v(K×)=Z. Such a sequence is called a saturated distinguished chain for α over K. We study how these chains are determined from α and see that these chains are closely related to the ramification of the field K(α).