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I want to calculate the irrational number, expressed by the following formula in gap: 3^(1/7). I've read through the related description here, but still can't figure out the trick. Will numbers like this appear in the computation of the character table and corresponding (unitary) representations?

P.S. Basically, I want to figure out the following question: For the computation of the character table and corresponding (unitary) representations, what is the minimum complete set of atomic irrational numbers used to express the results?

Regards, HZ

Hongyi Zhao
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You can't do that with GAP's standard cyclotomic numbers, as seventh roots of 3 are not cyclotomic. Indeed, suppose $r$ is such a root, i.e. a rot of the polynomial $f = x^7-3 \in \mathbb{Q}[x]$. Then $r$ is cyclotomic if and only if the field extension \mathbb{Q}[x] is a subfield of a cyclotomic field. By Kronecker-Weber this is equivalent to that field being an abelian extension, i.e., the Galois group is abelian. One can check that this is not the case here (the Galois group is a semidirect product of C_7 with C_6).

So, $r$ is not cyclotomic.

Max Horn
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  • Thank you for your in-depth explanation. As far as character table computation is concerned, which type of irrationalities will be used in all cases? Can the results always be expressed by cyclotomic numbers or other atomic irrationalities precisely/analytically without any approximation? From what I have seen so far, it seems that all the results in the feature table only need cyclotomic numbers. I'm not sure if this is related to the unitary representation. – Hongyi Zhao May 13 '22 at 15:21