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?



Discrete valuation sub-fields
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.


