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Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$.

  1. Does this module have a special name?
  2. Does a basis exist for every $n$? And if so, is there an algorithm to find a basis given an $n$?

I was just playing around with this, and noticed that for $n=3$ we have e.g. the basis $(1,\zeta)$ because $1+\zeta = -\zeta^2$. For $n=4$ we obviously have the basis $(1,i)$, but I was unable to generalize this for an arbitrary $n$.

Alex Wertheim
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flawr
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2 Answers2

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I don't know if you consider this a special name, but the ring $\mathbb{Z}[\zeta]$ is the ring of integers for the number field $\mathbb{Q}(\zeta)$. This is (IMO, anyway) a nontrivial fact, but you can find its proof in Neukirch (see below), or (for the case $n$ prime) in Samuel's ''Algebraic Theory of Numbers''—this link might also be helpful to you. Correspondingly, it is free as a $\mathbb{Z}$-module, and its rank is given by the degree $[\mathbb{Q}(\zeta):\mathbb{Q}] = \varphi(n)$, where $\varphi$ is the Euler totient function. The basis for $\mathbb{Z}[\zeta]$ as a $\mathbb{Z}$-module is given by $1, \zeta, \zeta^{2}, \ldots, \zeta^{d-1}$, where $d = \varphi(n)$. As a reference, see Neukirch's ''Algebraic Number Theory'', page 60, Proposition 10.2.

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Alex Wertheim
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  • Thank you very much, thats an excellent answer=) – flawr Jul 14 '16 at 18:59
  • @flawr: very glad I could help! :) – Alex Wertheim Jul 14 '16 at 18:59
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    Actually, OP did not ask for the ring of integers of $\Bbb Q(\zeta)$ — he asked only for the ring $\Bbb Z[\zeta]$. Isn’t it the case that if $\xi$ is integral over $\Bbb Z$, the module $\Bbb Z[\xi]$ is always $\Bbb Z$-free? After all, the module is finitely generated and torsion-free. – Lubin Jul 14 '16 at 19:54
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    @Lubin: you're absolutely right, so this answer is overkill. It seemed maybe that the OP was wondering what was special about the ring $\mathbb{Z}[\zeta]$, so this is what came to mind for me. :) – Alex Wertheim Jul 14 '16 at 19:57
  • Overkill maybe, but much more instructive than what I would have written. – Lubin Jul 14 '16 at 19:58
  • @Lubin I am still grateful that I learned about the context of number fields. Can you recommend any reference for learning module theory? This is a topic that I never really got in touch with. – flawr Jul 15 '16 at 10:31
  • This geezer is out of touch with the best modern literature. When I taught commutative algebra, I would have used Atiyah-MacDonald, but you’ll have to ask others than me. I think A-MacD is still available, but at a ridiculously high price. – Lubin Jul 15 '16 at 13:21
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The ring $\mathbb Z[\zeta]$ is called the ring of cyclotomic integers.

lhf
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  • Wikipedia and Mathworld both say that the term cyclotomic integers is only used for $\mathbb Z[\zeta]$ where $\zeta = e^{2\pi i/p}$ where $p$ is prime, do you agree? – flawr Jul 15 '16 at 10:14
  • @flawr, indeed, but I don't see why this should be. Other sources say that cyclotomic integers are the algebraic Integers in a cyclotomic field. See for instance, http://dx.doi.org/10.1016/0022-4049(81)90086-4 and http://www.ams.org/journals/mcom/1965-19-090/S0025-5718-1965-0180545-X/. – lhf Jul 15 '16 at 10:44