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.

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