Ternary cubic forms and étale algebras (Q2272255)

Summary: The configuration of inflection points on a nonsingular cubic curve in the complex projective plane has a well-known remarkable feature: the line by any two of the nine inflection points passes through a third inflection point. Therefore, the inflection points and the 12 lines through them form a tactical configuration \((94, 123)\), which is the configuration of points and lines of the affine plane over the field with 3 elements. This property was used by Hesse to show that the inflection points of a ternary cubic over the rationals are defined over a solvable extension. As a result, any ternary cubic can be brought to a normal form \(x^3_1 + x^3_2 + x^3_3-3x_1x_2x_3\) over a solvable extension of the base field. The purpose of this paper is to investigate this extension.