Good people!
I'm trying to prove a certain something, and I've reached a point where the whole thing will be complete if I can just prove the following lemma, which I'm actually not entirely certain is true. Hence I'm appealing to you for help!
Proposition:
Let $\mathbb{Q}(\alpha)$ be an algebraic extension of the field of rational numbers, and let this extension be of degree $n$. Then, there necessarily exists a primitive element $\beta \in \mathbb{Q}(\alpha)$ such that all algebraic integers in $\mathbb{Q}(\alpha)$ may be expressed in the form $$ v_0 + v_1 \beta + v_2 \beta^2 + \dots + v_{n-1} \beta^{n-1} , $$ where the $v_i$'s are all integers.
Is this true? If so, how does one go about proving it?
Is this false? If so, what would be a neat counterexample?
As always, thanks in advance.