Reciprocity law of finite Galois extension fields using Jacobian Varitey

TL;DR

This paper explores a reciprocity law for finite Galois extension fields of odd degree, using Jacobian varieties to embed roots into 2-torsion points and analyzing the Galois group's structure.

Contribution

It introduces a novel approach linking Galois extensions and Jacobian varieties, revealing the Galois group's subgroup structure within a general linear group over _{2}.

Findings

Roots embedded into 2-torsion points of Jacobian variety.

Galois group is a subgroup of GL(n, _{2}).

Reciprocity law established for specific Galois extensions.

Abstract

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 $\mathbb{F}_{2}$.