## Galois group of x^6-9x^6-9

f = x^6-9 = (x^3-3)(x^3+3) Let L_f be splitting field therefore L_f = \mathbb{Q}[\sqrt[3]{3},e^{\frac{2\pi i}{3}}], [L_f:\mathbb{Q}] = 9. Also Gal\space x^3±3/\mathbb{Q} = S_3 and Gal\space f/\mathbb{Q}-orbits is roots of x^3-3 and x^3+3 therefore, i think, that G = Gal\space f/\mathbb{Q} = S^3\times S^3. But |G| = 36 \ne [L_f:\mathbb{Q}]. What’s wrong? Answer A few issues: … Read more