Given a finite extension of the rationals, K, we know that K=Q[α] by the primitive element theorem, so every x∈K has the form

x=a0+a1α+⋯+anαn,

with ai∈Q.

However, the ring of integers, OK, of K need not have a basis over Z which consists of 1 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 that does not have a power basis be extended to a ring of integers OL which does have a power basis, for some finite L/K?

**Answer**

**Attribution***Source : Link , Question Author : Eins Null , Answer Author : Community*