Let be an algebraic extension of of degree and let be an integral basis. Let be the ring of integers in . Exhibit a nonzero rational integer and a basis (over ) for the ideal in such that, with , the matrix is upper triangular. (See Theorem 9.9 in TAN.)
Let , so that . Recall that is an integral basis for this . We saw in this previous exercise that is a basis for the ideal , and moreover has the desired property.
More generally, it is clear that is a basis for . The associated matrix is then not just triangular but diagonal.