# independence of valuations

Let $|\cdot|_{1}$, …, $|\cdot|_{n}$ be non-trivial (i.e., they all have also other values than 0 and 1) and pairwise non-equivalent valuations of a field $K$, all with values real numbers.  If $a_{1}$, …, $a_{n}$ are some elements of this field and $\varepsilon$ is an arbitrary positive number, then there exists in $K$ an element $y$ which satisfies the conditions

 $\displaystyle\begin{cases}|y-a_{1}|_{1}<\varepsilon,\\ \qquad\vdots\\ |y-a_{n}|_{n}<\varepsilon.\\ \end{cases}$
Title independence of valuations IndependenceOfValuations 2013-03-22 14:11:44 2013-03-22 14:11:44 pahio (2872) pahio (2872) 22 pahio (2872) Theorem msc 11R99 approximation theorem TrivialValuation EquivalentValuations WeakApproximationTheorem