Let be an algebraic extension of , and let be an algebraic integer. Prove that if is a unit, then the conjugates of are also units.
By Lemma 7.3, the norm is , where (by definition) is the product of the conjugates of . In particular, each conjugate of divides 1 (in the extended sense, meaning that is an algebraic integer). Thus each conjugate of is a unit.
Alternately, if is a unit then for some algebraic integer . Letting , , and , and letting be the conjugates of , then , where and denote the conjugates of and , respectively.