On the geometry of Galois cubic fields (Q650396)

The paper under review is concerned with studying the Galois fields over \(\mathbb{Q}\) of polynomials of the type \(f(x) = x^3 - kx +k\), where \(k\) is a positive integer, where \(f\) is irreducible over \(\mathbb{Q}\), and where the discriminant of \(f\) is a complete square. These conditions are satisfied precisely when \(k=n^2+n+7\) for some non-negative integer \(n\). After studying the Galois fields \(F_n\) of \(f(x) = x^3 -(n^2+n+7)x+n^2+n+7\) for \(n=0,1,\cdots,7\), and making several observations regarding their geometry, the author conjectures that some of these observations still hold true for other values of \(n\). The last section of the paper considers polynomials of the type \(x^3-kx+m\), where \(k\), \(m\) are distinct positive integers.