Timeline for Algebraic integers on the unit circle
Current License: CC BY-SA 2.5
11 events
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| Oct 1, 2010 at 15:35 | comment | added | Cam McLeman | Dirichlet's theorem says that the units in a number field are a free abelian group, except for a small little torsion piece consisting of roots of unity. The upshot is that the relations you describe are in fact the only ones. Dirichlet's Theorem is important and non-trivial -- it's worth picking up the algebraic overhead to really understand it. Any book titles "algebraic number theory" or related should do it. | |
| Oct 1, 2010 at 14:57 | comment | added | Vagabond | I found Dirichlet's unit theorem yesterday though being absolutely unfamiliar with algebra I was rather intimidated. Can you please recommend a reference, further if you think I should read some other results which you think may be useful please do mention. Once again many thanks for your patience. | |
| Oct 1, 2010 at 14:48 | comment | added | Vagabond | @btw you guessed right about being in a fixed number field. I also have one query, this field which is a finite extension of $\mathbb Q$ when I restrict it to the unit circle I will get a multiplicative subgroup say $H$, I can now try to quotient out $H$ by the previous group generated by $\theta_i's$, what do I get ? May be one does not get anything by studying these objects and its a blind alley but then how do I know ? So I asked. | |
| Oct 1, 2010 at 14:40 | comment | added | Cam McLeman | Right, though these non-trivial relations are themselves pretty trivial -- I'd spend some time with Dirichlet's Unit Theorem. | |
| Oct 1, 2010 at 14:32 | comment | added | Vagabond | more generally its possible that such a relation between the roots holds $\theta_1^{m_1} * \theta_2^{m_2} *\dots *\theta_k^{m_k} =1$. In that case certainly we wont get something like a free group as there are non trivial relation involving the roots ? | |
| Oct 1, 2010 at 14:26 | comment | added | Vagabond | So, I was thinking may be I should look into the group generated by these zeros which lie on the unit circle. Its quite possible that even if they are not roots of unity i.e. they d not satisfy $\theta^n=1$ for any n, a linear combination of them might do, for example if $\theta_1/\theta_2 = e^{2 \pi i k/n}$ then we get $(\theta_1 * \overline{ \theta_2})^n$ =1 | |
| Oct 1, 2010 at 14:20 | comment | added | Cam McLeman | The argument that the the roots lie on the unit circle is much simpler than Kronecker's Theorem. I'm sure it's in the first section or two of Washington's Cyclotomic Fields. Restricting to the irreducible case seems prudent -- otherwise you could take a cyclotomic polynomial and multiply it by any other monic polynomial and be unable to conclude much of anything. | |
| Oct 1, 2010 at 14:17 | history | edited | Cam McLeman | CC BY-SA 2.5 |
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| Oct 1, 2010 at 14:17 | comment | added | Vagabond | Thank you for your answer. I will try to elaborate on my question, hope it will clear some of the confusion. The reason why I was asking this question is the following. I have a monic-polynomial with integer coefficient, it may not be irreducible. I know that some of its root lies on the unit circle and my interest lie in finding some arithmetic/geometric relation between these zeros which lies in the unit circle. Now as you said by Kronecker theorem if all the roots lie on the unit circle then I can conclude that the roots are roots of unity. But I have no way to conclude that. | |
| Oct 1, 2010 at 13:40 | history | edited | Cam McLeman | CC BY-SA 2.5 |
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| Oct 1, 2010 at 13:35 | history | answered | Cam McLeman | CC BY-SA 2.5 |