Prove that if the Galois group of the splitting field of a cubic polynomial over is the cyclic group of order 3, then all the roots of the cubic polynomial are real.
How do I prove that the roots are real? Some hints please.
Yes. If there is a complex root then there is another complex root that is its conjugate. If you let be complex conjugation then is an automorphism of order 2. Which contradicts that the Galois group is cyclic of order 3 (therefore all its automorphism must have order dividing 3).