Principality of prime ideals of algebraic number fields

We discuss principality of prime ideals of finite algebraic number fields $L=K(\theta )$ over an algebraic number field $K([K:\mathbb{Q}]<\mathrm{\infty })$ defined by irreducible polynomials $f(x)\in {\mathfrak{O}}_K[x]$ and $f(\theta )=0$. Our main Theorem says that if a principal prime ideal $(\pi )\subset {\mathfrak{O}}_K$ is relatively prime to conductor $\mathfrak{F} ={\alpha\in \mathfrak{O}{L}|$aprincipalideal$(\alpha)$of$\mathfrak{O}{L}\subset \mathfrak{O}{K}[\theta]}$andsplitscompletelyover$L$:$(\pi)\mathfrak{O}{L}=\prod \mathfrak{p}{i}$,then$\mathfrak{p}{i}$ is a principal ideal of ${\mathfrak{O}}_L$ for all $i$, where $\mathfrak{O}_{L}= L \cap \overline{$\mathbb{Z}$}$isintegerringof$L$. We use Jacobian Varieties of non-singular projective curve model of super elliptic curves $y^l=f(x)$ to show the main Theorem, where $l$ is a large enough prime number which is relatively prime to degree of $f(x)$ and $(\pi )$.