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.
"Tschirnhausen transformation of a cubic generic polynomial and a $2$-dimensional involutive Cremona transformation." Proc. Japan Acad. Ser. A Math. Sci. 83 (3) 21 - 26, March 2007. https://doi.org/10.3792/pjaa.83.21