  We discuss principality of prime ideals of finite algebraic number fields
$L=K(\theta)$ over an algebraic number field $K ([K:\mathbb{Q}]<\infty)$
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 $\mathfrak{F}
=\{\alpha\in \mathfrak{O}_{L}|$ a principal ideal $(\alpha)$ of
$\mathfrak{O}_{L}\subset \mathfrak{O}_{K}[\theta]\}$ and splits completely over
$L$: $(\pi)\mathfrak{O}_{L}=\prod \mathfrak{p}_{i}$, then $\mathfrak{p}_{i}$ is
a principal ideal of $\mathfrak{O}_{L}$ for all $i$, where $\mathfrak{O}_{L}= L
\cap \overline{\mathbb{Z}}$ is integer ring of $L$. 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 $(\pi)$.
