Let be ideals in the ring of integers in , with . Show that if then , using only the existence of a basis for .
Let be a basis for over . ( exists by Theorem 7.10 since .) Since , we have for each , where , using the definition of the ideal product and collecting terms in as necessary. Rearranging, we have for each , where is the Kronecker delta. That is, is a nontrivial solution to the matrix equation , where . Thinking of as embedded in the field , we have . On the other hand, via the Leibniz expansion of , we have for some . So , and we have .