If $C$ is a characteristic $0$ algebraically closed field over $\mathbb{Q}_p$ which is complete with respect to a non-trivial non-archimedean valuation. Now let $x_1, \cdots,x_n$ be any $n$ elements of $C$. Is $x_1, \cdots,x_n$ contained in a sub-field of $C$ with discrete valuations?
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Not necessarily. Assuming $x\in C$ is an element with valuation $\sqrt{2}$, $x$ is not contained in any subfield of $C$ which has discrete valuation.
