Reciprocity law of finite Galois extension fields using Jacobian Varitey

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