In the context of field theory, let's say we are finding the minimal polynomial of the $\sqrt[3] 2$ over $\mathbb{Q}$.
Clearly a candidate will be $x^3-2$, which is irreducible by Eisenstein's criterion. Can we then immediately conclude that it is the minimal polynomial?
What I am worried is, is there such thing as an irreducible polynomial that is somehow not the minimal polynomial? (I know minimal polynomial implies irreducibility, but not sure about the converse: does irreducible polynomial implies minimality?)
Thanks for any help.