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 $(\pi )$.

arXiv commentBecause Theorem is not correct

References in corpus (1)

• Reciprocity law of finite Galois extension fields using Jacobian Varitey