A study of defectless and vs-defectless extensions of valued fields

Subject Areas : Commutative algebra

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

Received: 2021-03-14 Accepted : 2021-05-28 Published : 2021-06-01

Keywords:

References:

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

[2] K. Aghigh, S. K. Khanduja, On the main invariant of elements algebraic over a Henselian valued field, Proc. Edinb. Math. Soc. 45 (1) (2002), 219-227.

[3] K. Aghigh, A. Nikseresht, Characterizing distinguished pairs by using liftings of irreducible polynomials, Canad. Math. Bull. 58 (2) (2015), 225-232.

[4] K. Aghigh, A. Nikseresht, Constructing complete distinguished chains with given invariants, J. Algebra Appl. 14 (2015), 3:1550026.

[5] V. Alexandru, N. Popescu, A. Zaharescu, A theorem of characterization of residual transcendental extensions of a valuation, J. Math. Kyoto Univ. 28 (4) (1988), 579-592.

[6] J. Ax, Zeros of polynomials over local fields-the Galois action, J. Algebra. 15 (1970), 417-428.

[7] W. Baur, Die Theorie der Paare reell abgeschlossener Körper, (German) [The theory of pairs of real-closed fields] Logic and algorithmic (Zurich, 1980), pp. 25–34, Monograph. Enseign. Math., 30, Univ. Geneve, Geneva, 1982.

[8] S. Bhatia, S. K. Khanduja, On extensions generated by roots of lifting polynomials, Mathematika. 49 (2002), 107-118.

[9] A. Bishoni, S. K. Khanduja, On Eisenstein-Dumas and generalized Schönemann polynomials, Comm. Algebra. 38 (9) (2010), 3163-3173.

[10] A. Bishoni, S. Kumar, S. K. Khanduja, On liftings of powers of irreducible polynomials, J. Algebra Appl. 12 (2013), 5:1250222.

[11] R. Brown, Valuations, primes and irreducibility in polynomial rings and rational function fields, Trans. Amer. Math. Soc. 174 (1972), 451-488.

[12] R. Brown, Roots of irreducible polynomials in tame Henselian extension fields, Comm. Algebra. 37 (7) (2009), 2169-2183.

[13] R. Brown, J. L. Merzel, Invariants of defectless irreducible polynomials, J. Algebra Appl. 9 (4) (2010), 603-631.

[14] F. Delon, Extensions s'eparees et immediates de corps values. J. Symbolic Logic. 53 (2) (1988), 421-428.

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

[16] A. J. Engler, A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005.

[17] B. Green, M. Matignon, F. Pop, On valued function fields I, Manuscripta Math. 65 (3) (1989), 357-376.

[18] S. K. Khanduja, A note on a result of J. Ax, J. Algebra. 140 (1991), 360-361.

[19] S. K. Khanduja, R. Khassa, On invariants and strict systems of irreducible polynomials over Henselian valued fields, Comm. Algebra. 39 (2) (2011), 584-593.

[20] S. K. Khanduja, J. Saha, On a generalization of Eisenstein’s irreducibility criterion, Mathematika. 44 (1) (1997), 37-41.

[21] P. Kovacsics, F. V. Kuhlmann, A. Rzepka, On valuation independence and defectless extensions of valued fields, J. Algebra. 555 (2020), 69-95.

[22] F. V. Kuhlmann, Valuation Theory of Fields, Abelian Groups and Modules, Monograph in preparation, Preliminary versions of several chapters are available on the web site https://math.usask.ca/texttildelowfvk/Fvkbook.htm

[23] S. MacLane, A construction for absolute values on polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363-395.

[24] S. MacLane, A construction for prime ideals as absolute values of an algebraic field, Duke Math. J. 2 (1936), 492-510.

[25] S. MacLane, The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals, Trans. Amer. Math. Soc. 43 (1938), 226-239.

[26] N. Moraes de Oliveira, E. Nart, Defectless polynomials over henselian fields and inductive valuations, J. Algebra. 541 (2020), 270-307.

[27] K. Ota, On saturated distinguished chains over a local field, J. Number Theory. 79 (1999), 217-248.

[28] N. Popescu, A. Zaharescu, On the main invariant of an element over a local field, Portugal. Math. 54 (1) (1997), 73-83.

[29] N. Popescu, A. Zaharescu, On the structure of the irreducible polynomials over local fields, J. Number Theory. 52 (1) (1995), 98-118.

[30] O. F. G. Schilling, The Theory of Valuations, Amer. Math. Soc. Math. Surveys No. 4. Providence: Amer. Math. Soc. 1950.

[31] A. Singh, S. K. Khanduja, On construction of saturated distinguished chains, Mathematika. 54 (1-2) (2007), 59-65.

[32] A. Singh, S. K. Khanduja, On finite tame extensions of valued fields, Comm. Algebra. 33 (2005), 1095-1105

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