# Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $$KK$$, we know that $$K=Q[α]K=\mathbb{Q}[\alpha]$$ by the primitive element theorem, so every $$x∈Kx \in K$$ has the form

$$x=a0+a1α+⋯+anαn,x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$

with $$ai∈Qa_i \in \mathbb{Q}$$.

However, the ring of integers, $$OK\mathcal{O}_K$$, of $$KK$$ need not have a basis over $$Z\mathbb{Z}$$ which consists of $$11$$ and powers of a single element (a power basis). In fact, there exist number fields which require an arbitrarily large number of elements to form such a basis.

Question: Can every ring of integers $$OK\mathcal{O}_K$$ that does not have a power basis be extended to a ring of integers $$OL\mathcal{O}_L$$ which does have a power basis, for some finite $$L/KL/K$$?