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 \( \mathfrak{S} \)3 and C3 over k. As an application of the cubic case, we also give several sextic generic polynomials over k.2000 Mathematics Subject ClassificationPrimary 11R1612E2512F1012F1212F20Key words and phrasessubfield problemgeneric polynomialsTschirnhausen transformation

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