Khanduja, S.K.
(1992)
*On valuations of K(X)*
Proceedings of the Edinburgh Mathematical Society, 35
.
pp. 419-426.
ISSN 0013-0915

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## Abstract

For a valued field (K, v), let K_{v} denote the residue field of v and G_{v} its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any µin a totally ordered Abelian group containing G_{v}, and define a valuation w on K[x] by w(Σ_{i}c_{i}(x−α)^{i})=min_{i}.(ν
(c_{i})+iµ.Clearly either G_{v} is a subgroup of finite index in G_{W}=G_{v
}+ Zµ or G_{W}/G_{v} is not a torsion group. It can be easily shown that K(x)_{w} is a simple transcendental extension of K_{v} in the former case. Conversely it is well known that for an algebraically closed field K with a valuation ν, if w is an extension of ν to K(x) such that either K(x)_{w} is not algebraic over K_{v} or G_{w}/G_{v} is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

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ID Code: | 69878 |

Deposited On: | 17 Nov 2011 03:38 |

Last Modified: | 17 Nov 2011 03:38 |

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