Is it true that the units in $\mathbb{Q}[x,y]$ are precisely the constant polynomials?

Question

Is it true that the units in $\mathbb{Q}[x,y]$ are precisely the constant polynomials? I believe this is true for $\mathbb{Q}[x]$.

Answer 1

Hint: Try to prove: If a polynomial $$f=a_0+a_1x+\cdots+a_nx^n\in R[x]$$ is invertible, then $a_0$ is invertible and $a_i$ for $1\leq i\leq n$ is nilpotent.

Now suppose $\mathbb{Q}[x,y]$ as the ring $\mathbb{Q}[x][y]$.