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When one writes $\zeta_n$ which of the n roots of unity is meant here? Does it matter?

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up vote 10 down vote accepted

The context is important here.

It is rather standard for $\zeta_n$ to at least denote a primitive $n$th root of unity in the algebraic closure of the field $k$ which is currently being considered (this is possible iff the characteristic of $k$ does not divide $n$; in particular it is true for all fields of characteristic zero). When the characteristic of $k$ does not divide $n$ (i.e., when any primitive $n$th roots of unity exist) there are precisely $\varphi(n)$ primitive $n$th roots of unity, where $\varphi$ is Euler's phi function.

In a context in which $k$ is a subfield of the complex numbers, it is also rather standard for $\zeta_n$ to denote the specific primitive $n$th root of unity $e^{\frac{2 \pi i}{n}}$, i.e., the one of minimal argument in the complex plane.

Does it matter? For algebraic purposes, probably not: the primitive $n$th roots of unity in $\mathbb{C}$ are algebraically conjugate over $\mathbb{Q}$: i.e., different roots of a common irreducible polynomial over $\mathbb{Q}$, the cyclotomic polynomial $\Phi_n(t)$. Sometimes in number theory one considers systems of $n$th roots of unity for varying $n$, and in this case it is necessary to make a consistent (in a certain sense) choice of $\zeta_n$'s. Taking $\zeta_n = e^{\frac{2 \pi i}{n}}$ for all positive integers $n$ is such a consistent choice, but there are (many!) others.

Of course there are always situations when confusing one complex number for another would lead to trouble, so yes, in principle it might matter, especially in analytic or metric arguments.

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Even for algebraic purposes it can matter once you have several different roots of unity. For example, $\zeta_{20}+\zeta_{12}$ isn't well-defined even up to Galois conjugation. That is to say if you look at sums of different primitive 20th and 12th roots of unity there's more than one Galois orbit. – Noah Snyder Dec 8 '10 at 22:20
I am studying Euler's proof of Fermat's Last Theorem when the exponent is 3. The proof uses the notation $\zeta_3$. Im assuming this is either of the cube roots of unity that are not 1, since they are conjugates of each other. Is this correct? – Jason Smith Dec 8 '10 at 23:00
@Jason: sure, that's right. – Pete L. Clark Dec 9 '10 at 0:45

Usually this means $e^{ \frac{2\pi i}{n} }$. Depending on the context, it can just mean any primitive $n^{th}$ root of unity (and sometimes it doesn't matter which one, since they're all taken to each other under the action of the Galois group).

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Often when people write $\zeta_n$ as an nth root of unity, its to emphasize the fact that they aren't identifying it with a specific complex number. – Greg Muller Dec 8 '10 at 19:47

Probably a primitive one.

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