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.
Akinari Hoshi. Katsuya Miyake. "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