# Completeness of the field of fractions of a ring of formal power series

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.

• I think I have found the answer to my own question. Since I cannot post it here, I have posted it as an answer to the related question cited above. I hope it will be useful, since I have not been able to find it in any book I have had at hand. – Carlos Jan 20 at 15:02