Distinguished pairs of algebraic elements

Subject Areas : Statistics

َAzadeh Nikseresht ¹

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.

References:

[1] Aghigh K., Khanduja S.K., On chains associated with elements algebraic over a Henselian valued field, Algebra Colloq., 12 (4) (2005) 607-616.

 

[2] Anscombe, S., Kuhlmann, F.-V., Notes on extremal and tame valued fields, J. Symb. Log., 81 (2016) no. 2, 400-416.

 

[3] Azgin S., Kuhlmann F.-V., Pop F., Characterization of extremal valued fields, Proc. Amer. Math. Soc., 140 (2012) no. 5, 1535-1547.

 

[4] Blaszczok, A., Distances of elements in valued field extensions, Manuscripta Math., 159 (2019) no. 3-4, 397-429.

 

[5] Kuhlmann F.-V., A classification of Artin-Schreier defect extensions and characterizations of defectless fields, Illinois J. Math., 54 (2010) no. 2, 397-448.

 

[6] Lang S., Algebra, revised third ed., Addison-Wesley Publishing Company Advanced Book Program, Reading, MA (2002).

 

[7] Blaszczok, A., Infinite towers of Artin–Schreier defect extensions of rational function fields. In: Second International Conference and Workshop on Valuation Theory (Segovia/El Escorial, Spain, 2011), EMS Series of Congress Reports, 10 (2014) 16-54.

 

[8] Blaszczok, A., Kuhlmann, F.-V., Counting of the number of distinct distances of elements in valued field extensions. J. Algebra, 509 (2018) 192-211. 

 

[9] Ershov, Yu.L., Extremal valued fields, Algebra i Logika 43 (2004) 582-588, 631. English translation: Algebra and Logic, 43 (2004) 327-330.

 

[10] Ershov, Yu.L., *-extremal valued fields, Sibirsk. Mat. Zh., 50 (2009) 1280-1284.

 

[11] Endler O., Valuation Theory, Springer-Verlag, New York-Heidelberg, Berlin (1972).

 

[12] Engler A.J., Prestel A., Valued Fields, Springer-Verlag, Berlin (2005).

 

[13] Zariski O., Samuel P., Commutative Algebra, Vol. II. Springer-Verlag, New York-Heidelberg (1975).

 

[14] Popescu N., Zaharescu A., On the structure of the irreducible polynomials over local fields, J. Number Theory, 52 (1995) 98-118.

 

[15] Khanduja S.K., Saha J., A generalized fundamental principle, Mathematika, 46 (1999) 83-92.

 

[16] Aghigh K., Khanduja S.K., On the main invariant of elements algebraic over a Henselian valued field, Proc. Edinb. Math. Soc., 45 (2002) no. 1, 219-227.

 

[17] Brown R., Merzel J.L., Invariants of defectless irreducible polynomials, J. Algebra Appl., 9 (2010) no. 4, 603-631.

 

[18] Aghigh K., Nikseresht A., Characterizing distinguished pairs by using liftings of irreducible polynomials, Canad. Math. Bull., 58 (2015) no. 2, 225-232.

 

[19] Aghigh  K., Nikseresht A., Constructing complete distinguished chains with given invariants, J. Algebra Appl., 14 (2015) no. 3, 1550026, 10 pp.