The action of a Galois group on a cyclotomic field.

Question

Let $\zeta_5$ be a primitive $5^{th}$ root of unity, $k=\mathbb{Q}(\sqrt[5]n,\zeta_5)$ the normal closure of $\Gamma = \mathbb{Q}(\sqrt[5]n)$, ${\rm Gal}(k/\Gamma)= <\tau>$ with $\tau : \zeta_5 \longrightarrow \zeta_5^4$ and $\tau : \sqrt[5]n \longrightarrow \sqrt[5]n$ . If $n=5^ep^{e_1}$ with $p\equiv 1 [5]$, its known that $p$ split in the $5^{th}$ cyclotomic field $\mathbb{Q}(\zeta_5)$ as: $p=\pi_1\pi_2\pi_3\pi_4$ with $\pi_i$ are primes in $\mathbb{Q}(\zeta_5)$, my need to know what is the action of $\tau$ on $\pi_i$ in this case ?