Tschirnhausen transformation of a cubic generic polynomial and a $2$-dimensional involutive Cremona transformation

Abstract

We study the field isomorphism problem for a cubic generic polynomial $X^3+sX+s$ via Tschirnhausen transformation. Through this process, there naturally appears a $2$-dimensional involutive Cremona transformation. We show that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.