I am reading a textbook prior to taking my first course in Field Theory. I think that if someone could answer the following 4 questions simply as True of False I might be less confused. I am denoting the 3rd root of unity as $\zeta_3$.
1) Does $\mathbb Q(\sqrt[3]2)$ contain ALL the roots of the polynomial $x^3 - 2$?
2) Does $\mathbb Q(\zeta_3\sqrt[3]2)$ contain ALL the roots of the polynomial $x^3 - 2$?
3) Does $\mathbb Q(\sqrt[3]2,\zeta_3)$ contain ALL the roots of the polynomial $x^3 - 2$?
4) Is $\mathbb Q(\sqrt[3]2,\zeta_3)$ the SMALLEST field that contains ALL the roots of the polynomial $x^3 - 2$?
If the answers are False, False, True, True respectively then I think I am OK. Otherwise some gentle explanation would help me.