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Let $n$ be a positive integer and let $Q_n^d$ be the set of algebraic integers $\zeta$ which live in a degree $d$ extension of the rationals and such that any Galois-conjugate $\zeta^\sigma$ of $\zeta$ has complex norm $n$. The set $Q_n^d$ is finite since the coefficients of the minimal polynomial of any such $\zeta$ are bounded. Moreover it is not difficult to show that if $n = 1$ the set $S$ contains only roots of unity.

What is known about $Q_n^d$ ? For example, is it possible to list all its elements for small values of $n$ ?

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  • $\begingroup$ Since, as you note, it's a finite set, it is certainly possible to list all its elements for any given values of $n$ and $d$. Just look at all the (finitely many) irreducible polynomials of degree dividing $d$ and meeting the coefficient bound, and see which ones work. So, you must be asking something else --- but, what? $\endgroup$ Jun 13, 2015 at 5:46
  • $\begingroup$ For what it's worth, if $n$ is a prime power $p^r$, these are known as "$p$-Weil numbers of weight 2r" and they're very important in the study of motives, Galois representations, etc. I think it's a fair question to ask if there's a reasonable algorithm for listing the elements of $Q^d_n$ efficiently, particularly since it's not entirely trivial to recognise whether a given algebraic number is in $Q^d_n$ or not. $\endgroup$ Jun 13, 2015 at 8:18
  • $\begingroup$ Thank you very much for the answer. I have just downloaded some texts about $p$-Weil numbers. $\endgroup$
    – bog
    Jun 14, 2015 at 1:09

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