Timeline for Cyclic cubic numbers as rational linear combinations of roots of unity

5 events

when toggle format	what		by	license	comment
Jan 22, 2012 at 22:22	comment	added	Will Sawin		Yes, since the maxmimal real subfield of the field generated by $e^{2\pi i k/n)$ is the one generated by $\cos ( 2 \pi k/n)$.
Jan 22, 2012 at 22:00	comment	added	Wolfgang		I wonder if the roots (i.e. real numbers) that can be written as an explicit rational linear combination of roots of unity are exactly those that can be written as an explicit rational linear combination of trig functions of rational angles (which I call "nice roots").
Jan 22, 2012 at 21:57	comment	added	Wolfgang		At least for this case, the polynomials are also "nice" in the sense I have just defined in the MO thread mathoverflow.net/questions/86401/…. E.g., for $a=1$ we can write the roots as $2(\cos\frac{k\pi}{13}+\cos\frac{5k\pi}{13}),\ k=1,3,9$. for $a=2$ as $1-2 (\cos\frac{k\pi}{19}+\cos\frac{7k\pi}{19}+\cos\frac{11k\pi}{19}),\ k=1,3,9$, for $a=3$ as $1-\sqrt{12}\sin\frac{k\pi}{9},\ k=1,2,4$, etc.
Jan 4, 2012 at 10:52	comment	added	Chandan Singh Dalawat		That's a nice special case. Thanks for the reference.
Jan 4, 2012 at 10:29	history	answered	Chua KS	CC BY-SA 3.0