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The motivation of the question is that I try to test when a real number is not an cyclotomic integers. Or more specifically, when a positive real number is not a quantum dimension of a unitary fusion category?

We know that when $1\leq d<2$, $d$ is not a quantum dimension of a unitary fusion category if $d \neq 2\cos(\pi/n), \ n=3,4,5,\cdots$

One possible answer is an efficient algorithm to find an approximation of a number in terms of a cyclotomic integer. Just like there is an efficient algorithm to find an approximation of a real number in terms of a rational number.

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    $\begingroup$ What sort of answer are you expecting? A positive real number is an algebraic integer if and only if it is a root of a monic polynomial with integer coefficients. $\endgroup$ – Geoff Robinson Oct 26 '15 at 19:45
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    $\begingroup$ It sounds like you might be interested in arxiv.org/abs/1004.0665 , as well as many of the other recent papers of Scott Morrison and Noah Snyder front.math.ucdavis.edu/… $\endgroup$ – David E Speyer Oct 26 '15 at 23:24
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    $\begingroup$ A real number $x$ is an algebraic integer if and only if $-x$ is. So positivity plays no role. $\endgroup$ – P Vanchinathan Oct 27 '15 at 2:27
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    $\begingroup$ Well, $\mathbb Z+\sqrt2\mathbb Z$ is a set of cyclotomic integers dense in $\mathbb R$, hence you can approximate arbitrary reals by those. $\endgroup$ – Emil Jeřábek Oct 27 '15 at 14:26
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    $\begingroup$ Addressing all of the comments above: Xiao-Gang didn't realize he meant to ask about totally real cyclotomic integers. With this, it becomes quite interesting. $\endgroup$ – Scott Morrison Oct 28 '15 at 4:35
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I suspect you might be looking for the following fact:

If $x$ is a cyclotomic integer, and $p$ a prime does not divide the discriminant, then the minimal polynomial of $x$ factors modulo $p$ into irreducible components all of the same degree.

See for example Theorem 4.6 in Elementary and analytic theory of algebraic numbers by Władysław Narkiewicz.

In practice, this very effectively detects algebraic integers which are not cyclotomic, as used for example in Algorithm 3.6 of David Penneys and James E. Tener, Subfactors of index less than 5, Part 4: Vines, Internat. J. Math. 23 (2012), no. 3, 1250017, 18.

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  • $\begingroup$ When we start with an irreducible polynomial over some field and regards its factorization into irreducible in a Galois extension are not the degrees of irreducible factors above equal (this follows from the transitive action of the Galois group on the primes lying above a given prime: Here the dedekind domains are the PIDs given by polynomial rings over the base and extension fields). So I don't fully understand what is special about cyclotomy here. $\endgroup$ – P Vanchinathan Oct 28 '15 at 6:06
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    $\begingroup$ @P Vanchinathan: I don't understand the relevance of your comment. Here we are asking about the factorization of the minimal polynomial of $x$ over $\mathbb{F}_p$, which isn't a nontrivial Galois extension of anything. In general the degrees of such factorizations (for $p$ not dividing the discriminant) correspond to cycle types in the Galois group of the polynomial acting on its roots, by the Frobenius density theorem. In particular they are usually different. $\endgroup$ – Qiaochu Yuan Oct 28 '15 at 7:10
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    $\begingroup$ "Subfactors of index less than 5" is a very nice sequence of papers that contain many things I need to constrain quantum dimensions. So I select this as an answer. The one by @Dave Penneys is also very much related but I cannot choose two. $\endgroup$ – Xiao-Gang Wen Oct 30 '15 at 17:57
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Since you are looking for restrictions on quantum dimensions of objects in unitary fusion categories, you also want your cyclotomic integers to be totally real, as they are Frobenius-Perron eigenvalues of finite graphs. The recent article of Calegari-Guo http://arxiv.org/pdf/1502.00035v1.pdf (see Proposition 4.3 in particular) extends the results in the Calegari-Morrison-Snyder article referenced above by @David Speyer.

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  • $\begingroup$ Only 2 out of 5 possible quantum dimensions in Thm 1.01 in arxiv.org/pdf/1004.0665v1.pdf appear in the table on page 6 of arxiv.org/pdf/1502.00035v1.pdf. Does this mean the table on page 6 is not complete (so it cannot be used as necessary conditions)? $\endgroup$ – Xiao-Gang Wen Oct 28 '15 at 2:12
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    $\begingroup$ The table of page 6 of the Calegari-Guo paper is a complete list of totally real cyclotomic integers $\beta$ subject to the additional condition $\mathcal{M}(\beta) < 14/5$. $\endgroup$ – Scott Morrison Oct 28 '15 at 4:22

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