The set of atomic irrational numbers used to express the character table and corresponding (unitary) representations

Question

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

Answer 1

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.