arXiv subjects
Publications and source records attributed to Shinji Ishida.
In this note, we treat two dimensional complete local rings which are called "semi-stable local rings" and discuss the tame class filed theory. In 1987, Professor S. Saito completed the unramified class field theory of the general two dimensional excellent local rings. On the other hand, we discuss the tame class field theory for a specific local rings.
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 $(\pi)$.
For finite Galois extension fields defined by odd degree irreducible polynomials over algebraic integer ring, we observe "Reciprocity Law" through Jacobian Variety by embedding all roots of the polynomials into 2-torsion points of Jacobian Variety. Furthermore, Galois group of the minimal splitting field of such a polynomial is a subgroup of general linear group with coefficient $\mathbb{F}_{2}$.
We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} + \cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root $α_{n}$ of $f_{1,n}(x)$ satisfies $|α_{n}|\to 1$ ($n\to \infty$), and discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ for several type $n$. After that, we show that the Galois group of $f_{1,n}(x)$ is Symmetric group $S_{n-1}$ for several type $n$. Although those roots of $f_{1,n}(x)=0$ don't draw an exact circle, it looks like a circle on complex plane. Moreover by considering that Galois groups of $f_{1,n}(x)$ are not abelian in many cases, we call such extension fields over $\mathbb{Q}$ "Non-Abelian Cycrotomic Fields" here.