On valuations of K(X)

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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For a valued field (K, v), let Kv denote the residue field of v and Gv 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 Gv, and define a valuation w on K[x] by w(Σici(x−α)i)=mini.(ν (ci)+iµ.Clearly either Gv is a subgroup of finite index in GW=Gv + Zµ or GW/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv 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 Kv or Gw/Gv 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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