Exhibit a basis for an ideal in an algebraic integer ring which is upper triangular in an appropriate sense

Let $K$ be an algebraic extension of $\mathbb{Q}$ of degree $n$ and let $\{ \omega_i\}$ be an integral basis. Let $\mathcal{O}$ be the ring of integers in $K$. Exhibit a nonzero rational integer $k$ and a basis (over $\mathbb{Z}$) $\{\alpha_i\}$ for the ideal $(k)$ in $\mathcal{O}$ such that, with $\alpha_i = \sum c_{i,j}\omega_j$, the matrix $[c_{i,j}]$ is upper triangular. (See Theorem 9.9 in TAN.)

Let $K = \mathbb{Q}(i)$, so that $\mathcal{O} = \mathbb{Z}[i]$. Recall that $\{1,i\}$ is an integral basis for this $K$. We saw in this previous exercise that $\{2,2i\}$ is a basis for the ideal $(2)$, and moreover has the desired property.

More generally, it is clear that $\{k\omega_i\}$ is a basis for $(k)$. The associated matrix is then not just triangular but diagonal.