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- W2085103486 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 μ in 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 ( v ( 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 v , if w is an extension of v 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." @default.
- W2085103486 created "2016-06-24" @default.
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- W2085103486 date "1992-10-01" @default.
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- W2085103486 title "On valuations of K(x)" @default.
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- W2085103486 doi "https://doi.org/10.1017/s0013091500005708" @default.
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