Let be an algebraic number field with ring of integers , and let be an ideal. Suppose such that . Prove that .
Let be a basis for over . (This exists by Theorem 7.10). In particular, for each , there exist rational integers such that . Rearranging, we have for each , where is the Kronecker delta. In particular, is a nontrivial solution to the matrix equation , where . Thus . On the other hand, by the Leibniz expansion for determinants, is a polynomial in having coefficients in . Thus is an algebraic integer, and more specifically .