Let be an algebraic number field and let be an integral basis for . Let be a nonzero algebraic integer. Show that is a basis of an ideal of integers in if and only if .
Suppose first that is a basis of an ideal – that is, . Certainly, since , we have . Now is an integral basis for , so that in particular there exist rational integers such that . Then , and we have . Hence .
Conversely, suppose . Certainly , where denotes the set of all -linear combinations of elements from . Again because is an integral basis for , we have . Hence . Next we claim that is -linearly independent. To that end, suppose for some ; then since the ring of integers in is an integral domain, . Since is an integral basis for , we have . Thus is -linearly independent, and so is a basis for .