Let $k$ be a field and let $k[[X,Y]]$ be the ring of formal power series with coefficients in $k$. Let $k((X,Y))$ be its field of fractions. For $F\in k[[X,Y]]$, $F\neq 0$ define $v(F)$ as the least degree of a monomial appearing in $F$, and extend $v$ to $k((X,Y))$ by $v(F/G)=v(F)-v(G)$. Then $v$ is a valuation in $k((X,Y))$ that induces an absolute value (namely $|F|=2^{-v(F)}$) and it is well-known that $k[[X,Y]]$ is a complete ring, but:
is the field of fractions $k((X,Y))$ also complete?
This is a particular case of this question.
With just one indeterminate the answer is positive.
