Suppose $p$ be an odd prime such that $p \equiv 1 \pmod 4$. In the ring of Gaussian integers $\mathbb{Z}[i]$, $p$ factors as $p = v \cdot \overline{v}$ for a prime $v \in \mathbb{Z}[i].$
Prove that $v$ and $\overline{v}$ are not associates. i.e. prove that the two ideals $(v)$ and $(\overline{v})$ are distinct.
If $v = a + bi,$ then direct computation yields: $ v \cdot 1 \ne \overline{v}, \hspace{1mm} v \cdot -1 \ne \overline{v},\hspace{1mm} v \cdot i \ne \overline{v}, \hspace{1mm}v \cdot -i \ne \overline{v}.$ Thus $v$ and $\overline{v}$ are not associates.
$($an explicit calculation would be $v \cdot -i = (a+bi) \cdot -i = b - ai \ne a - bi)$
Am I missing any details?