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Tschirnhausen transformation of a cubic generic polynomial and a \(2\)-dimensional involutive Cremona transformation. (English) Zbl 1126.14018

The authors 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. The authors prove that the fixed field under the action of the transformation is purely transcendental over an arbitrary base field.

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
12F12 Inverse Galois theory

Software:

Mathematica
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References:

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