Distinguished pairs of algebraic elements

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Nikseresht َAzadeh

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Journal Paper

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Journal: JOURNAL OF NEW RESEARCHES IN MATHEMATICS Year:2022 | Volume:8 | Issue:37 Page(s): 197-208

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Information Journal Paper

Title

Distinguished pairs of algebraic elements

Author(s)

Nikseresht َAzadeh | Issue Writer Certificate

Keywords

extensions of valuations

algebraic elements over a valued field

distinguished pair

minimal polynomial

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 element M(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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APA: Copy

Nikseresht, َAzadeh. (2022). Distinguished pairs of algebraic elements. JOURNAL OF NEW RESEARCHES IN MATHEMATICS, 8(37 ), 197-208. SID. https://sid.ir/paper/1065019/en

Vancouver: Copy

Nikseresht َAzadeh. Distinguished pairs of algebraic elements. JOURNAL OF NEW RESEARCHES IN MATHEMATICS[Internet]. 2022;8(37 ):197-208. Available from: https://sid.ir/paper/1065019/en

IEEE: Copy

َAzadeh Nikseresht, “Distinguished pairs of algebraic elements,” JOURNAL OF NEW RESEARCHES IN MATHEMATICS, vol. 8, no. 37 , pp. 197–208, 2022, [Online]. Available: https://sid.ir/paper/1065019/en

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