D+F 7.3.2

Prove that the rings $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.

$\textit{Proof}$ : In order for $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ to be isomorphic, they must have exactly the same algebraic structure. Now, the units of $\mathbb{Z}$ are $$\mathbb{Z}^*=\{1,-1\}$$ The units of $\mathbb{Q}$ are $$\mathbb{Q}^*=\mathbb{Q}-\{0\}$$ By proposition 4, p. 235, the above are the units of $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$, respectively. With completely different subsets of units, $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ cannot be isomorphic. $\blacksquare$