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    About on distinguished chains
    (MTJPAM Turkey, 2024) Öztürk B.; Öke F.
    Let v be a valuation of a field K and a = a0, a1, …, am be a complete distinguished chain for a ? ¯K\K. Let va be the extension of the valuation v to the field K(a). In this study some properties provided by distinguished pairs and lifting polynomials used to define a residual algebraic free extension of v to the rational function field K(x) are obtained. The value group and residue field of the valuation va are written with the help of constants ?K (ai) and minimal polynomials of the elements in the chain. In addition to the relations between value groups and residue fields of the valuations vai bi of the fields K(ai, bi) are studied by considering two chains a = a0, a1, …, am and b = b0, b1, …, bm. © 2024, MTJPAM Turkey. All rights reserved.
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    Some constants and tame extensions according to a valuation of a field with rank v = 2
    (2012) Öztürk B.; Öke F.
    Let v = v1 ° v2 be a valuation of a field K with rankv = 2 and v? be the extension of v to the algebraic closure K? of K. Let (L, z)/(K, v) be a finite extension of valued fields where z = z 1 ° z2 be the extension of v to field L. In this paper it is shown that, if (L, z)/(K, v) be a tame extension then finite extensions of valued fields (L, z1)/(K, v1) and (kz1, z2)/(kv1, v2) are tame extensions. Also Krasner's constant of an element ? ? K?\K is obtained as w (k, v) (?) = (w(K, v1) (?), w(kv1, v2) (?*)) and the other constants of ? are obtained as ?(K, v) (?) = (?(K, V1) (?), ?(kv1, v2) (?*)) and ?(K, v) (?) = (?(K,v1) (?), ?(kv1 V1)(?*)).