Principality of prime ideals of algebraic number fields

Abstract

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 $(\pi)$.