## A conjugate of a unit is a unit

Let $K$ be an algebraic extension of $\mathbb{Q}$, and let $u \in K$ be an algebraic integer. Prove that if $u$ is a unit, then the conjugates of $u$ are also units.

By Lemma 7.3, the norm $N(u)$ is $\pm 1$, where (by definition) $N(u)$ is the product of the conjugates of $u$. In particular, each conjugate of $u$ divides 1 (in the extended sense, meaning that $1/u^\prime$ is an algebraic integer). Thus each conjugate of $u$ is a unit.

Alternately, if $u$ is a unit then $uv = 1$ for some algebraic integer $v$. Letting $K = \mathbb{Q}(\theta)$, $u = r(\theta)$, and $v = s(\theta)$, and letting $\theta_i$ be the conjugates of $\theta$, then $u_iv_i = r(\theta_i)s(\theta_i) = 1$, where $u_i$ and $v_i$ denote the conjugates of $u$ and $v$, respectively.