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 $K$.
[Math] Fraction field of the formal power series ring in finitely many variables
commutative-algebra
Best 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.