Let $z \in \mathbb{C}$ be an algebraic integer. Is then $|z| \in \mathbb{R}$ also an algebraic integer?
2026-03-25 07:35:31.1774424131
Modules of algebraic integers
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With $z$ also $\overline{z}$ is an algebraic integer, because it is a root of the same minimal polynomial. Since the algebraic integers form a ring, also the product $z\cdot\overline{z}$ is an algebraic integer. Then also its square root, i.e., $|z|=\sqrt{z\overline{z}}$.