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.
We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.

Mathematics Subject Classification: 03C60, 12J10

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Full text arXiv 1308.3394: pdf, ps.