Let $\Gamma = \mathbb{Q}(\sqrt[5]n)$ be a purely quintic field with $n = 5^ep^{e_1}$ and $p$ a prime number verify $p\equiv 1 [5]$ and $p\not\equiv 1 [25]$.
Let $k_0 = \mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic field. It is known that $k=\mathbb{Q}(\sqrt[5]n,\zeta_5)$ is the normal closure of $\Gamma$, and $Gal(k/Q) = \langle \sigma,\tau\rangle$ with $Gal(k/\Gamma) = \langle\tau\rangle$, $Gal(k/k_0) = \langle\sigma\rangle$ and $\tau^4 = 1$, $\sigma^5 = 1$. We know that $p$ split in $k_0$ as $p=\pi_1\pi_2\pi_3\pi_4$ with $\pi_i$ are prime in $k_0$.
I have these problems :
- If $p\not\equiv 1 [25]$ we can deduce that all $\pi_i\not\equiv 1 [25]$? Or is there any information about the congruence of the prime $\pi_i$?
- What is the action of $\tau$ on the primes of $k_0$?
- What is explicitly the genus field of the extension $k/k_0$?