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  1 - Distinguished pairs of algebraic elements
  َAzadeh Nikseresht
  10.30495/jnrm.2022.60572.2086
  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 ̃( More

  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. Manuscript profile