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Preprint Number 615
615. Krzysztof Krupinski Superrosy fields and valuations E-mail: Submission date: 15 August 2013. Abstract: We prove that every non-trivial valuation on an infinite superrosy field of
positive characteristic has divisible value group and algebraically closed
residue field. In fact, we prove the following more general result. Let K be
a field such that for every finite extension L of K and for every natural
number n>0 the index [L^*:(L^*)^n] is finite and, if char(K)=p>0 and f:
L \to L is given by f(x)=x^p-x, the index [L^+:f[L]] is also finite. Then
either there is a non-trivial definable valuation on K, or every non-trivial
valuation on K has divisible value group and, if char(K)>0, it has
algebraically closed residue field. In the zero characteristic case, we get
some partial results of this kind. Mathematics Subject Classification: 03C60, 12J10 Keywords and phrases: |

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