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Distinguished pairs of algebraic elements | ||
| Journal of New Researches in Mathematics | ||
| مقاله 10، دوره 8، شماره 37، آذر و دی 2022، صفحه 197-208 اصل مقاله (312.24 K) | ||
| نوع مقاله: research paper | ||
| شناسه دیجیتال (DOI): 10.30495/jnrm.2022.60572.2086 | ||
| نویسنده | ||
َAzadeh Nikseresht  
| ||
| ,Department of Mathematics, ,Ayatollah Boroujerdi University, Boroujerd,,Iran. | ||
| چکیده | ||
| 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 degα>degβ 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. | ||
| کلیدواژهها | ||
| extensions of valuations؛ algebraic elements over a valued field؛ distinguished pair؛ minimal polynomial | ||
| مراجع | ||
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