Whether $\mathbb{Q}(\alpha,\beta,\gamma,\delta,\eta)\subseteq\mathbb{Q}(e^{2\pi i/n})$ or not?

Asked Aug 01 '24 at 04:36

Active Aug 01 '24 at 05:16

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I am wondering if $\mathbb{Q}(\alpha,\beta,\gamma,\delta,\eta)\subseteq\mathbb{Q}(e^{2\pi i/n})$ for some integer $n$ ?

where

$\alpha=\frac{1+\sqrt{17}}{2}$

$\beta=\frac{1-\sqrt{17}}{2}$

$\gamma=-1+2\cos\left(\frac{\pi}{9}\right)$

$\delta=-1-\cos\left(\frac{\pi}{9}\right)+\sqrt{3}\sin\left(\frac{\pi}{9}\right)$

$\eta=-1-\cos\left(\frac{\pi}{9}\right)-\sqrt{3}\sin\left(\frac{\pi}{9}\right)$

These special values come from the roots of characteristic polynomial $$(x-3)(x-2)^{18}x^{17}(x^2-x-4)^9(x^3+3x^2-3)^{16}$$ of the Biggs–Smith graph. It seems hard to determine if it is in the very large cyclotomic field.

edited Aug 01 '24 at 05:16

asked Aug 01 '24 at 04:36

user1992

1,536

2

Yes, we can take $n = 17 \cdot 9 = 153$. For $\alpha$ and $\beta$ this follows from standard facts about Gauss sums and for the other three it follows from those closed forms. – Qiaochu Yuan Aug 01 '24 at 05:22
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Those cosines and sines surely belong to a cyclotomic field. Also all square roots belong to some cyclotomic field (an instance of Kronecker-Weber, but invoking that is overkill). A compositum of cyclotomic fields is cyclotomic itself. – Jyrki Lahtonen Aug 01 '24 at 05:24
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Would this thread cover everything you want to know? – Jyrki Lahtonen Aug 01 '24 at 05:26
@JyrkiLahtonen: Yeah, very appreciate your help. – user1992 Aug 01 '24 at 05:51
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There's also this oldie. But I'm fairly sure Qiaochu Yuan gave you the smallest $n$ that works :-) – Jyrki Lahtonen Aug 01 '24 at 06:05
You can greatly simplify, $u=\sqrt{17},v=\cos(\pi/9), w=\sqrt2\sin(\pi/9).$ So you want an $n$ with $@,b,c\in\mathbb Q[e^{2\pi/n}].$ – Thomas Andrews Aug 01 '24 at 06:33

Whether $\mathbb{Q}(\alpha,\beta,\gamma,\delta,\eta)\subseteq\mathbb{Q}(e^{2\pi i/n})$ or not?

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