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Let $a$ be an algebraic integer of degree $n$ and $K$ is the number field $\mathbb{Q}(a)$ with $\mathcal{O}_K$ as its ring of integers. Let $\sigma_1,\dots,\sigma_n$ be the embeddings of $K$.

I want to show (or disprove) that given $M>0$, the set of $x\in\mathcal{O}_K$ such that $|\sigma_i(x)|\leq M$ for all $i$ is finite.

I thought of using the norm of $x$ but I am stuck.

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    https://math.stackexchange.com/questions/92426/how-many-elements-in-a-number-field-of-a-given-norm – Eric Towers Dec 09 '20 at 03:24
  • @EricTowers The duplicate is incorrect. @ morphy22 The set is finite because $x$ is a root of some polynomial $\in \Bbb{Z}[X]$ with coefficients $\le 2^n M^n$. – reuns Dec 09 '20 at 09:25
  • @reuns: Sorry. I had to work on some other things. How can I show this? –  Mar 22 '21 at 20:57
  • The $\sigma_j(x)$ are the roots of some degree $n$ monic polynomial with integer coefficients, use $M$ to bound the coefficients to get a finite set of possible polynomials. – reuns Mar 22 '21 at 21:13

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