Power basis for a number field

Question

In our class today we stated a Proposition that if $ E/F $ is a finite separable extension, $ F $ is the fraction field of $ A $ and $ B $ is the integral closure of $ A $ in $ E $, then $ B $ is a free $ A $-module and any basis for $ B $ over $ A $ is a basis for $ E $ over $ F $. A quick remark was made that bases of the form $ (1, \alpha, \dots, \alpha^{n-1}) $ don't always exist. Of the (admittedly few) cases I have seen so far, there is always a power basis. What is an example of a situation where power bases don't exist?

Answer 1

I will give an example of number field. In your notation, let $F=\mathbb{Q}$, $E=\mathbb{Q}(\sqrt{-2},\sqrt{-5})$, $A=\mathbb{Z}$. Then the ring of integers $B$ in $E$ cannot be written the form $\mathbb{Z}[\zeta]$ for any $\zeta\in B$. You can try to prove this by finding an integral basis. Although it will be quite computational without some theorems.

This property of algebraic number field is called monogenity.