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What is the fraction field of the formal power series ring over a field in finitely many variables $K[[X_1,\dots,X_n]]$? Is there a nice description for this field? When $n=1$, I know this is the formal Laurent series ring over $K$.

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    This is a good question. In order to get something similar to the case $n=1$ it would be interesting to know if a formal power series in several indeterminates can be written as a product between a polynomial from $(X_1,\dots,X_n)$ and an invertible power series. (I don't know if this is true or not.)2013-02-14
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    @YACP You're asking if $K[[X_1,\dots,X_n]] = K[[X_1,\dots, X_n]]^{\times} K[X_1,\dots,X_n]$?2013-02-14
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    And this is false for $n\ge 2$. Consider $y^2-x^2(1+x)$ which is irreducible in $\mathbb C[x,y]$. It is the product of $y-x\sqrt{1+x}$ and of $y+x\sqrt{1+x}$ as power series. If the latter are polynomials $P, Q\in (x,y)\mathbb C[x,y]$ up to units, then $y^2-x^2(1+x)$ and $PQ$ generate the same ideal in $\mathbb C[[x,y]]$. By faithfull flatness, both ideals would be equal in $\mathbb C[x,y]$, which is impossible2013-02-14

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