On value groups and residue fields of some valued function fields

Khanduja, Sudesh K. (1994) On value groups and residue fields of some valued function fields Proceedings of the Edinburgh Mathematical Society, 37 . pp. 445-454. ISSN 0013-0915

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Let K = K0(x,y) be a function field of transcendence degree one over a field K0 with x,y satisfying y2 = F(x), F(x) being any polynomial over K0. Let v0 be a valuation of K0 having a residue field K0 and v be a prolongation of v0 to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of v0 and v, then either G/G0 is a torsion group or there exists an (explicitly constructive) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element ϒ of G such that G is the direct sum of G1 and the cyclic group Zϒ. As regards the residue fields, a method of explicitly determining A: has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math. 65 (1989), 357-376], \_Manuscripta Math. 58 (1987), 179-214]).

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