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Mathematics B‑class Mid‑priority

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if n is not divisible by the characteristic of the field

My wording about exactly n nth roots was a bit reckless, but I did not claim that z = 1 always has distinct roots. BTW I just learned about a (weaker) definition of cyclotomically closed field, which apparently is exactly what we need here. Incnis Mrsi (talk) 18:03, 5 April 2013 (UTC)

Geometry

It would be nice to have a "Geometric interpretation" section that describes the fact that all n roots of unity lie at equidistant points on the unit circle (centered at the origin) in the cartesian/complex plane, i.e., they lie on the vertices of a regular n-gon inscribed within the unit circle. But I don't know where the most appropriate place is to add such a section in the article. — Loadmaster (talk) 19:00, 30 September 2013 (UTC)

yes. what is the meaning of this? it seems special 24.98.133.72 (talk) 01:39, 19 August 2014 (UTC)

Well, multiplying any combination of the nth roots of 1 always produces one of the (other) roots (or 1) on the complex unit circle. Does that qualify as "special"? There is no deeper meaning, since these are just some of the properties of the unit multiplier (1) in the complex field. — Loadmaster (talk) 19:15, 19 August 2014 (UTC)

Positive power?

Random observation while I was passing through: should the opening paragraph specify that n is positive? After all, raising any complex number (save 0) to the 0th power yields 1. 0 is obviously an integer, but that doesn't mean that the number is a root of unity. (The definition below does a better job, specifying that n must be positive.)

Aasmith (talk) 06:34, 16 September 2015 (UTC)

 Fixed D.Lazard (talk) 08:14, 16 September 2015 (UTC)

I hate do not understanding stuff!

In this case, what I don't understand is this assertion in the article:

Generally, z ∈ R can be considered for any field R

I'm sure this is a stupid question*, but does this mean that in general, a complex number is a real? I guess it depends on the definition of "considered."

Even if not, why talk about complex numbers that are points on the real number line? The imaginary part is zero, so it's a degenerative, trivial complex number.

Right?

For someone who hates not understanding stuff, I sure do it a lot.

__________________

• I can't stand when people preface questions with that when asking me, but now I see why they do. --Verdana♥Bold 07:57, 4 March 2017 (UTC)

You are confusing R, a variable, which, here, denotes an arbitrary ring, with the standard symbol for the field of real numbers, which is ${\displaystyle \mathbb {R} }$ or R. Nevertheless, I agree that the formulation was confusing, and not only because of the use of R (which was unnecessary). I have thus rewritten the paragraph. D.Lazard (talk) 08:57, 4 March 2017 (UTC)

Roots of Unity Filter

Useful for generating functions http://zacharyabel.com/papers/Multi-GF_A06_MathRefl.pdf Wqwt (talk) 22:32, 1 September 2018 (UTC)

This cannot be used in WP, as being original research, not regularly published. Moreover, this result fails to satisfy the WP:notability criteria, because of the lack of WP:secondary sources. D.Lazard (talk) 06:52, 2 September 2018 (UTC)

If you consider brilliant.org a valid source then it shows up there too. I'm sure it's in some textbook somewhere, not only for contest math, but probably under a different name. It makes an appearance in Problems from the Book by Titu Andreescu. Wqwt (talk) 02:10, 3 September 2018 (UTC)

Merge Root of unity modulo n

I think there's enough overlap between this article and Root of unity modulo n to warrant merging it with this article. Wqwt (talk) 01:30, 3 September 2018 (UTC)

• Oppose. Although related to Root of unity and Finite field#Roots of unity, this article is about a different case: the roots of unity in a ring that is not a field. I have added a link to Root of unity modulo n in the body of Root of unity. Nevertheless Root of unity modulo n would require to be cleaned up by adding context, explicit examples, and better explanation of the differences with the case of roots of unity in a field. D.Lazard (talk) 10:13, 2 October 2018 (UTC)
• better not! Jackzhp (talk) 13:19, 10 October 2018 (UTC)
• I agree that this seems like a bad idea, and will remove the tag. --JBL (talk) 23:52, 12 January 2019 (UTC)

Recent edits

Since September, 28, Xayahrainie43 made 25 edits to the articles. A few of them introduce minor improvements, but most do not improve the article and consist mainly of introducing data for roots of unity from 9 to 12. Here are more details on these additions:

• Section "Cyclotomic polynomials" is intended for being a summary of the article of the same name (template {{main article}}. So adding details for these degrees makes no sense: Either they duplicate the content of the main article or, if they are useful, they should be added to the main article.
• In section "Algebraic expressions", no explicit formula is given in the edits. There are only general unsourced comments, that either duplicate the beginning of the section, or are not understandable for most readers (references without explanation to casus irreducibilis, or even are completely wrong (I had to revert twice the wrong assertion that eleventh roots of unity cannot expressed in radicals.

Thus I'll revert all these edits, and, when useful, I'll add a few words for clarification. Please, do not add data for higher degree, without getting a consensus here (see WP:CONSENSUS and WP:BRD). D.Lazard (talk) 16:20, 29 September 2018 (UTC)

In the first illustration ...

... leave off the x- and y-axes!!! Just show the roots of unity — as the caption states.2600:1700:E1C0:F340:38C4:1D3E:151:63CC (talk) 03:25, 2 October 2018 (UTC)

It makes no sense to show the roots of unity without showing where they are in the complex plane. However it may useful to clarify the caption by saying that the roots are the blue dots. This has been done. D.Lazard (talk) 08:35, 2 October 2018 (UTC)

Unclear statement

As the fourth bullet point in the section Explicit expressions in low degrees, this passage appears:

"*As Φ₅(x) = x + x + x + x + 1, the four primitive fifth roots of unity are the roots of this quartic polynomial, which may be explicitly solved in terms of radicals, giving the roots

${\displaystyle {\frac {\varepsilon {\sqrt {5}}-1}{4}}\pm i{\frac {\sqrt {10+2\varepsilon {\sqrt {5}}}}{4}},}$

where ${\displaystyle \varepsilon }$ may take the two values 1 and –1."

But it needs to be stated (if true) that the two instances of ε are independent — or not, rather than left to the reader to wonder which is the case.2600:1700:E1C0:F340:38C4:1D3E:151:63CC (talk) 03:34, 2 October 2018 (UTC)

 Fixed (You could have done this edit yourself.) D.Lazard (talk) 08:37, 2 October 2018 (UTC)

Primitive root

IMHO nobody—except for Trente percente—cares about such technicalities. The term primitive root is explained in this article, not elsewhere. Hence it should be boldfaced one time. Incnis Mrsi (talk) 09:31, 9 September 2019 (UTC)

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