Principal ideal by considering a product How can I show by considering the product of $\zeta_7^3 + \zeta_7^2 +1$ with $\zeta_7^3 + \zeta_7 +1$ that $(2, \zeta_7^3 + \zeta_7^2 +1 )$ is a principal ideal of $O_{Q(\zeta_7)}$?

Let $\zeta=\zeta_7$. Multiply the numbers together. Doing so gives you the following:
\begin{align*}
(\zeta^3+\zeta^2+1)(\zeta^3+\zeta+1)&=\zeta^6+\zeta^5+\zeta^3+\zeta^4+\zeta^3+\zeta+\zeta^3+\zeta^2+1\\&=(\zeta^6+\zeta^5+\zeta^4+\zeta^3+\zeta^2+\zeta+1)+2\zeta^3\\
&=2\zeta^3
\end{align*}
$\zeta$ is a unit in this ring, so $2\in (\zeta^3+\zeta^2+1)$. In other words, $(2,\zeta^3+\zeta^2+1)=(\zeta^3+\zeta^2+1)$.
