 Solution

Start with any generator $I=(\beta )$ and use the decomposition from part (d). For each $i$, choose an integer $n_i$ such thatand putSince $u$ is a unit, $(\alpha )=(\beta )=I$. Moreover,The vector $h$ lies in the fixed compact parallelepipedHence there is a constant $M_K$ such that every real coordinate of every $h\in P$ is at most $M_K$, and every complex coordinate is at most $2M_K$.

For a real embedding ${\sigma }_i$,For a chosen complex embedding ${\tau }_j$, the weighted coordinate givesThe conjugate embedding has the same modulus. Exponentiating and taking any $C>e^{M_K}$ proves the archimedean balancing of a principal ideal generator:for every embedding $\sigma :K\to \mathbb{C}$, with $C$ depending only on $K$.