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

2007/10/01 by Akinari Hoshi, Hoshi, Akinari, Katsuya Miyake +1
Mathematics · #11R16 #12E25 #12F10 #12F12 #12F20 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11R16 #msc:12E25 #msc:12F10 #msc:12F12 #msc:12F20

paper · pdf · doi:10.48550/arxiv.0710.0287

38 pages, added some corollaries

openalex publication_date 2007/10/01 · arxiv created 2008/10/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \frakS₃ and C₃ over k. As an application of the cubic case, we also give several sextic generic polynomials over k.

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