Hilbert's Irreducibility Theorem and Ideal Class Groups of Quadratic Fields

TL;DR

This paper proves a quadratic version of Hilbert's Irreducibility Theorem, providing quantitative improvements for counting quadratic fields with specific ideal class groups, based on binary form value results.

Contribution

It introduces a quadratic case of Hilbert's Irreducibility Theorem with enhanced quantitative bounds and applies it to ideal class group counting in quadratic fields.

Findings

Quantitative bounds for quadratic Hilbert's Irreducibility Theorem

Improved counts of quadratic fields with certain ideal class groups

Application of Stewart and Top's results on binary forms

Abstract

We prove a version of Hilbert's Irreducibility Theorem in the quadratic case, giving a quantitative improvement to a result of Bilu-Gillibert in this restricted setting. As an application, we give improvements to several quantitative results counting quadratic fields with certain types of ideal class groups. The proof of the main theorem is based on a result of Stewart and Top on values of binary forms modulo squares.