Counting Eisenstein polynomials satisfying a condition from genus theory

TL;DR

This paper provides an asymptotic count of monic Eisenstein polynomials of odd prime degree that meet a specific condition related to the genus number of an algebraic number field.

Contribution

It introduces a new asymptotic formula for counting Eisenstein polynomials with a condition linked to genus theory, expanding understanding in algebraic number theory.

Findings

Derived an asymptotic formula for the count of such polynomials

Connected polynomial counting to genus number conditions

Enhanced understanding of polynomial distribution in algebraic number fields

Abstract

We give an asymptotic formula for the number of monic Eisenstein polynomials of odd prime degree satisfying an additional condition that arises in the study of the genus number of an algebraic number field.