A geometric framework for the subfield problem of generic polynomials via Tschirnhausen transformation

TL;DR

This paper introduces a geometric framework using Tschirnhausen transformations to solve subfield problems of generic polynomials, providing explicit solutions for cubic cases and constructing sextic polynomials.

Contribution

It develops a general geometric method for subfield problems of generic polynomials and explicitly solves the cubic case, extending to sextic polynomials.

Findings

Explicit solutions for cubic generic polynomials over arbitrary fields.

Construction of several sextic generic polynomials.

A unified geometric approach to subfield problems.

Abstract

Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for $\frak{S}_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$.