I'm stuck on the following seemingly simple exercise:

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> Let $K$ be a number field and $\eta\in K$ an algebraic *integer*. If for every complex embedding $\varphi$ of $K$ we have $|\varphi(\eta)|=1$, then $\eta$ is a root of unity.
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I am able to show that (if $\eta\neq1$) the minimal polynomial of $\eta$ over $\Bbb{Q}$ is palindromic, so in particular its constant term is $1$ and $\eta$ is a unit in $\Bbb{Z}[\eta]$. Any hint for where to look further is appreciated.

Also, I'm somewhat surprised by the requirement that $\eta$ be an algebraic integer, and wonder whether it is necessary. Any insight here is also welcome.