Distinguished pairs of algebraic elements

Subject Areas : Statistics

َAzadeh  Nikseresht ¹ (,Department of Mathematics, ,Ayatollah Boroujerdi University, Boroujerd,,Iran.)

Received: 2021-05-07 Accepted : 2022-05-14 Published : 2022-08-23

Keywords: چندجمله‌ای مینیمال, توسیع‌های ارزیاب, جفت متمایز, عناصر جبری روی میدان ارزیاب,

Abstract :

Let v be a henselian valuation on a field K, and v ̃ be its unique extension to the algebraic closure K ̃ of K. An element α∈K ̃K has a distinguished pair if the corresponding set M(α,K) (defined as the following) has a maximum elementM(α,K)={v ̃(α-β)┤β in K ̃,[K(β) ∶K]<[K(α) ∶K]}.In this case, a pair (α,β) of elements of K ̃ is a distinguished pair for α whenever β is an element of smallest degree over K such that deg⁡α>deg⁡β and v ̃(α-β)=supM(α,K). In this paper, we first present some results about distinguished pairs of algebraic elements of arbitrary degree over henselian valued fields. Then considering the importance of algebraic elements of prime degree in the extensions of valued fields, we concentrate on such elements. In particular, for α∈K ̃ of prime degree over K, we give a necessary and sufficient condition for the existence of the maximum of the corresponding set M(α,K) by using the minimal polynomial of α over K.

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