Is the dimension of a finitely generated $K$-subalgebra of $K[X_1,\ldots,X_n]$ bounded above by $n$?

Question

Let $K$ be a field. Is there an example of a finitely generated $K$-subalgebra $$ A\subseteq K[X_1,\ldots, X_n] $$ of Krull dimension $\dim A>n$? If yes, is there such an example for $n=1$?

Answer 1

This is not possible.

The Krull dimension of a finitely generated $K$-algebra $A$ which is an integral domain equals the transcendence degree of $Q(A)$ (the field of fractions of $A$) over $K$. Since $Q(A)\subset K(X_1,\dots,X_n)$ and the transcendence degree of $K(X_1,\dots,X_n)$ over $K$ is $n$ we are done.