I'm doing an introductory course of field theory and there is one excercise that as easy as it seems it bring me on my nerves. It states:
Let $α_1, \dots, α_n ∈ \mathbb{C}$ be the roots of an irreducible polynomial $f ∈ Q[x]$ of degree $n ≥ 2$.
(a) Prove that there exist infinitely many tuples $(c_2,\dots , c_n) ∈ \mathbb{Z}^{n−1}$ such that $α_1 + c_2α_2 + \dots+ c_nα_n$ is a primitive element of the splitting field of $f$ over $Q$.
(b) Check that $α_1 + α_2 + \dots + α_n$ is never a primitive element of the splitting field of $f$ over $Q$.