On the field intersection problem of quartic generic polynomials via formal Tschirnhausen transformation

TL;DR

This paper addresses the field intersection problem for quartic generic polynomials over fields of characteristic not 2, using formal Tschirnhausen transformations and multi-resolvent polynomials to find solutions.

Contribution

It introduces a novel approach combining formal Tschirnhausen transformations with multi-resolvent polynomials to solve the field intersection problem for quartic generic polynomials.

Findings

Solved the field intersection problem for quartic generic polynomials

Developed a method using formal Tschirnhausen transformations

Applied multi-resolvent polynomials to analyze polynomial intersections

Abstract

Let $k$ be a field of characteristic $\neq 2$. We give an answer to the field intersection problem of quartic generic polynomials over $k$ via formal Tschirnhausen transformation and multi-resolvent polynomials.