Q:

Is $K[X_1,X_2,...]$ a G-domain?

Recall that an integral domain $D$ with quotient field $K$ is called a Goldman domain (or G-domain) if $K$ is finitely generated ring over $D$.
Is $K[X_1,X_2,...]$ a G-domain, where $K$ is a field ?

A:

We know that if $R$ is an integral domain then the polynomial ring $R[X]$ is not a Goldman domain; see Proposition 12.5.
We can then view
$$ K[X_1,X_2,\ldots] $$
as
$$K[X_2,\dots][X_1].$$

