Timeline for Finite order elements of $\mathrm{GL}_d(\mathbb{Z})$ that are conjugate to powers of themselves
Current License: CC BY-SA 4.0
8 events
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| Feb 4, 2021 at 10:37 | comment | added | YCor | @GeoffRobinson Oh, yes, I focussed on $A$ being conjugate to $A^{-1}$ after Alex B.'s answer. So, I should have said that the set of roots of $\Phi_n$ is stable under $x\mapsto x^m$ for $m$ coprime to $n$. | |
| Feb 4, 2021 at 10:29 | comment | added | Geoff Robinson | @YCor : What you say is true, but I think you need a little more than stable under inversion: the roots of irreducible cyclotomic polynomials are all Galois conjugate to each other (Galois groups being over $\mathbb{Q}$). | |
| Feb 4, 2021 at 10:24 | comment | added | YCor | Just as a motivation, if I'm correct, this is always true in $\mathrm{GL}_d(\mathbf{Q})$, due to the fact that the set of roots of every cyclotomic polynomial is stable under inversion. (And this is clearly false in $\mathrm{GL}_d(\mathbf{C})$ for every $d\ge 1$, just use a scalar matrix of finite order $\ge 3$.) | |
| Feb 4, 2021 at 0:27 | vote | accept | Caleb Eckhardt | ||
| Feb 4, 2021 at 0:00 | history | edited | YCor | CC BY-SA 4.0 |
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| Feb 3, 2021 at 23:55 | history | edited | YCor |
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| Feb 3, 2021 at 23:20 | answer | added | Alex B. | timeline score: 14 | |
| Feb 3, 2021 at 22:14 | history | asked | Caleb Eckhardt | CC BY-SA 4.0 |