Criterion for the Galois Group of an Irreducible Cubic over an Arbitrary Field Suppose K is a field and f x x ax bx c K x is irreducible so the Galois group of f x over K is either S3 or A3 a Show that the Galois group of f x is A3 if and only if the resultant quadratic poly nomial g x x ab 3c x b a c 6abc 9c has a root in K If a B y are the roots of f x show that the Galois group is A3 if and only if the element 0 a By ya is an element of K and that is a root of g x Show that the discriminant of g x is the same as the discriminant of f x b ch K 2 If K has characteristic different from 2 show either from a or directly from the definition of the discriminant that the Galois group of f x is A3 if and only if the discriminant of f x is a square in K c ch K 2 If K has characteristic 2 show that the discriminant of f x is always a square Show that f x can be taken to be of the form x px q and that the Galois group of f x is A3 if and only if the quadratic x qx p q has a root in K equivalently if p3 q2 q2 K is in the image of the Artin Schreier map xx x mapping K to K d If K F 1 where is transcendental over F Prove that the polynomials x 1 x 1 x 1 1 1 x 1 1 and x3 1 1 x 1 1 1 have A3 as Galois group while x3 x t and x x t have S3 as Galois group