Abstract
For valued fields K of rank higher than 1, we describe how elements in the henselization K h of K can be approximated from within K; our result is a handy generalization of the well-known fact that in rank 1, all of these elements lie in the completion of K. We apply the result to show that if an element z algebraic over K can be approximated from within K in the same way as an element in K h, then K(z) is not linearly disjoint from K h over K.
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Kuhlmann, FV. Approximation of elements in henselizations. manuscripta math. 136, 461–474 (2011). https://doi.org/10.1007/s00229-011-0453-x
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DOI: https://doi.org/10.1007/s00229-011-0453-x