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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## Abstract

Let K = K_{0}(x,y) be a function field of transcendence degree one over a field K_{0} with x,y satisfying y^{2} = F(x),
F(x) being any polynomial over K_{0}. Let v_{0} be a valuation of K_{0} having a residue field K_{0} and v be a
prolongation of v_{0} to K with residue field k. In the present paper, it is proved that if G_{0}⊆G are the value
groups of v_{0} and v, then either G/G_{0} is a torsion group or there exists an (explicitly constructive) subgroup
G_{1} of G containing G_{0} with [G_{1}:G_{0}]<∞ together with an element ϒ of G such that G is the direct sum of G_{1}
and the cyclic group Z_{ϒ}. As regards the residue fields, a method of explicitly determining A: has been described
in case k/k_{0} is a non-algebraic extension and char k_{0}≠2. The description leads to an inequality relating the
genus of K/K_{0} with that of k/k_{0}: 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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ID Code: | 69896 |

Deposited On: | 17 Nov 2011 03:40 |

Last Modified: | 17 Nov 2011 03:40 |

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