Let $Gamma = mathbb{Q}(sqrt[3]n)$ a pure cubic subject and $p$ prime in $mathbb{Z}$. Now we have the next theorem launched by Dedekind:

Theorem: (1) If $p$ divides $n$ and $pneq3$ so $p$ is ramified in $Gamma$, then $pmathcal{O}_Gamma$ = $mathcal{P}^3$ and $mathcal{N}(mathcal{P}) = p$

(2) if $pnmid n$ and $pequiv -1(mod, 3)$, so $p$ is unramified in $Gamma$ then : $pmathcal{O}_Gamma$ = $mathcal{P}mathcal{P}_1$ and $mathcal{N}(mathcal{P}) = p$, $mathcal{N}(mathcal{P}_1) = p^2$

(3) if $pnmid n$ and $pequiv 1(mod, 3)$, so $p$ is unramified in $Gamma$ then :

• $pmathcal{O}_Gamma$ = $mathcal{P}mathcal{P}_1mathcal{P}_2$ and $mathcal{N}(mathcal{P}) = mathcal{N}(mathcal{P}_1)=mathcal{N}(mathcal{P}_2)$ if $n$ is a cubic residue modulo $p$

• $pmathcal{O}_Gamma$ = $mathcal{P}$ and $mathcal{N}(mathcal{P}) = p^3$ if $n$ is just not a cubic residue modulo $p$

the decomposition within the pure cubic subject is solved by this theorem, my query is concerning the pure quintic subject $mathbb{Q}(sqrt[5]n)$, for the primary level of the concept we’ve the identical outcome for ramified prime, however for the unramified prime I have to now what’s their decomposition