When one creates $\zeta_n$ which of the n origins of unity is suggested below?

When one creates $\zeta_n$ which of the n origins of unity is suggested below? Does it matter?

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2019-12-02 02:51:36
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Answers: 3

Probably a primitive one.

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2019-12-03 04:20:50
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Usually this suggests $e^{ \frac{2\pi i}{n} }$. Relying on the context, it can simply suggest any kind of primitive $n^{th}$ origin of unity (and also occasionally no matter which one, given that they are all taken to each various other under the activity of the Galois team).

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2019-12-03 04:20:41
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The context is necessary below.

It is instead typical for $\zeta_n$ to at the very least represent a primitive $n$th origin of unity in the algebraic closure of the area $k$ which is presently being taken into consideration (this is feasible iff the feature of $k$ does not separate $n$ ; specifically it holds true for all areas of particular absolutely no). When the feature of $k$ does not separate $n$ (i.e., when any kind of primitive $n$th origins of unity exist) there are specifically $\varphi(n)$ primitive $n$th origins of unity, where $\varphi$ is Euler is phi function.

In a context in which $k$ is a subfield of the complex numbers, it is additionally instead typical for $\zeta_n$ to represent the details primitive $n$th origin of unity $e^{\frac{2 \pi i}{n}}$, i.e., the among marginal argument in the facility aircraft.

Does it matter? For algebraic objectives, possibly not : the primitive $n$th origins of unity in $\mathbb{C}$ are algebraically conjugate over $\mathbb{Q}$ : i.e., various origins of an usual irreducible polynomial over $\mathbb{Q}$, the cyclotomic polynomial $\Phi_n(t)$. Occasionally in number theory one takes into consideration systems of $n$th origins of unity for differing $n$, and also in this instance it is essential to make a regular (in a particular feeling) selection of $\zeta_n$'s. Taking $\zeta_n = e^{\frac{2 \pi i}{n}}$ for all favorable integers $n$ is such a regular selection, yet there are (several!) others.

Certainly there are constantly scenarios when perplexing one intricate number for an additional would certainly bring about problem, so of course, in concept it could matter, specifically in analytic or statistics debates.

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2019-12-03 04:14:10
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