Exhibit a basis for an algebraic extension which consists of integers but which is not integral.
Suppose is a squarefree integer congruent to 1 mod 4 and consider . Certainly is a basis of this field extension of . Moreover, both 1 and are algebraic integers (trivially and via , respectively).
However, note that is an algebraic integer, since it is a root of . It is not, however, a -linear combination of . So this basis is not integral.