Let $k = \mathbb{Q}(\zeta_5)$ the cyclotomic field, where $\zeta_5$ a primitive $5^{th}$ root of unity.
let $p$ a prime integer such that $p \equiv 1 [5]$, more exactly $p \equiv 1 [25]$, then $p$ decompose in $\mathcal{O}_{k}$ as: $p=\pi_1\pi_2\pi_3\pi_4$, where $\pi_i$ are primes in $\mathcal{O}_{k}$.
If $\pi$ is a prime of $\mathcal{O}_{k}$ and $\alpha\in \mathcal{O}_{k}$, we define the quintic power residue symbol $(\frac{\pi}{\alpha})$ to be the fifth root of unity such that : $(\frac{\pi}{\alpha}) \equiv \alpha^{\frac{N(\pi)-1}{5}} [\pi]$. Where N is the absolute norm.
My question is: For the prime $p \equiv 1 [25]$ such that $p=\pi_1\pi_2\pi_3\pi_4$, I need to prove that $(\frac{\pi_i}{\pi_j}) =1$ for $i,j \in \{1,2,3,4\}$ and $i\neq j$.