Approximation of elements in henselizations

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For valued fields K of rank higher than 1, we describe how elements in the henselization Kʰ 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 Kʰ, then K(z) is not linearly disjoint from Kʰ over K.