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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.) –  user26857 Feb 14 '13 at 17:01
    
@YACP You're asking if $K[[X_1,\dots,X_n]] = K[[X_1,\dots, X_n]]^{\times} K[X_1,\dots,X_n]$? –  JSchlather Feb 14 '13 at 18:04
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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 impossible –  user18119 Feb 14 '13 at 21:55
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1 Answer

There’s a really big difference between the one-variable power series ring, together with its fraction field, and a many-variable power series ring. Namely that the one-variable ring has only one irreducible element. This is what lets us have a nice description of the fraction field of $k[[x]]$. A many-variable ring, even $k[[x,y]]$, has infinitely many irreducibles unrelated by unit factors. Much uglier, from this viewpoint, than the one-variable case.

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Is there an infinite class of irreducible elements in $k[[x,y]]$ that are easy to describe? –  JSchlather Feb 14 '13 at 19:38
    
@Jacob, this is so far from my competence that I hesitate to answer. Although there are polynomials in $k[x,y]$ that split in $k[[x,y]]$, I wonder whether every associate class of irreducibles in the power series ring has an irreducible polynomial in it. –  Lubin Feb 15 '13 at 18:28
    
@Jacob: sorry, my “I wonder whether” was completely stupid. Clearly, $y−f_1(x)$ and $y−f_2(x)$, where $f_1\ne f_2$, are two irreducible power series in $k[[x,y]]$ whose ratio is not a unit of the power-series ring. –  Lubin Feb 15 '13 at 19:30
    
Dear Lubin, how can you convince us that power series are far from you competence ? –  user18119 Feb 15 '13 at 22:14
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