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I'm stuck on the following seemingly simple exercise:

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.

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.

Servaes
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  • It's a theorem of Kronecker. See, e.g., https://math.stackexchange.com/questions/2121231/a-question-of-the-proof-of-kronecker-s-theorem-regarding-cyclotomic-polynomials and https://math.stackexchange.com/questions/4323/are-all-algebraic-integers-with-absolute-value-1-roots-of-unity and https://math.stackexchange.com/questions/1731038/algebraic-number-with-conjugates-having-modulus-1 or just type something like Kronecker theorem root of unity into Google. – Gerry Myerson Oct 27 '19 at 01:05

2 Answers2

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There is this argument which I found in a book by Hecke, but may be older:

Hint: show that there are only finitely many algebraic integers in $K$ all of whose conjugates have absolute value $1$.

Indeed, the coefficients of the minimal polynomial of $\eta$ are bounded in terms of $[K : \mathbb Q]$: the coefficient of $x^k$ is bounded by $\binom {[K : \mathbb Q]}k$. Thus $\eta$ belongs to the set of roots of a polynomial which belongs to a finite set. Hence the cyclic group generated by $\eta$ is finite.

Bart Michels
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$\mathcal O_K\subset K$ is a lattice, and thus the image of $\mathcal O_K$ in $\Bbb C^n$ (where $n=[K:\Bbb Q]$) via all embeddings $\phi:K\to\Bbb C$ is discrete and closed. The image of the set $\{\eta^n|n\in\Bbb Z\}$ is thus discrete closed and bounded, so it is finite, and thus $\eta^n=1$ for some $n>0$.

As to why algebraic integer is needed - consider $(1+2i)/(1-2i)$.

user8268
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