Is the ring of formal power series $\mathbb K[[x_1, \dots, X_n]]$ in $n$ indeterminates over a field a PID?

Question

Is the local ring of formal power series $\mathbb K[[x_1, \dots, X_n]]$ in $n$ indeterminates $x_1, \dots, x_n$, $n>1$, over a field $\mathbb K$ a principal ideal domain (PID)? This is true when $n=1$, but is it true for $n>1$? Thanks in advance.

Answer 1

A quick answer using the following well-known dimension theory result:

"If $A$ is a commutative noetherian ring, then $$\dim A[[x]] = \dim A +1."$$

Arguing by induction, it follows that $$\dim \mathbb{K}[[x_1,\dots,x_n]]=n$$ for every $n \geq 0$.

However, if $A$ is PID, then $\dim A \leq 1$, so we conclude that $\mathbb{K}[[x_1,\dots,x_n]]$ is a PID if and only if $n=0$ or $n=1$.