Exercise 15.4.9

Proof. The ring $\mathbb{Z}[i]$ is principal, thus is a UFD with field of fractions $\mathbb{Q}[i]$. By Gauss’s Lemma (Theorem A.5.8), if $P,Q\in \mathbb{Z}[i][u]$ are relatively prime in $\mathbb{Z}[i][u]$, then $P,Q$ are relatively prime in $\mathbb{Q}[i][u]$.

Since $P_{\beta }(u),Q_{\beta }(u)$ are relatively prime in $\mathbb{Q}[i][u]$, there are some polynomials $A,B\in \mathbb{Q}[i][u]$ such that $A(u)P_{\beta }(u)+B(u)Q_{\beta }(u)=1$. Reasoning by contradiction, suppose that $u{P}_{\beta }(u)$ and $Q_{\beta }(u)$ have a common root $\alpha$ in $ℂ$. Since $Q_{\beta }(0)=1$, $\alpha \ne 0$, thus $P_{\beta }(\alpha )=0$. Then $P_{\beta }(\alpha )=Q_{\beta }(\alpha )=0$ implies $1=A(\alpha )P_{\beta }(\alpha )+B(\alpha )Q_{\beta }(\alpha )=0$: this is a contradiction.

Thus $u{P}_{\beta }(u)$ and $Q_{\beta }(u)$ have no common roots in $ℂ$. □