Principality of prime ideals of algebraic number fields

TL;DR

This paper investigates the principality of prime ideals in algebraic number fields, establishing conditions under which prime ideals are principal when splitting completely, using Jacobian varieties of super elliptic curves.

Contribution

It proves a new theorem relating prime ideal principality to splitting behavior and uses Jacobian varieties of super elliptic curves to demonstrate the result.

Findings

Prime ideals splitting completely are principal under certain conditions.

Use of Jacobian varieties of super elliptic curves in number theory.

Main theorem connects ideal principality with splitting and curve models.

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 $(\pi)$.