Talk:Root of unity

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)

Explicit Expression of the 7th Root of unity

The real part of the 7th roots of unity can be expressed as the solutions of 8x³ + 4x² - 4x - 1 = 0, which can be overlaid with the unit circle to find all the roots. As is, the method to find the seventh roots is notably more complicated than this. — Preceding unsigned comment added by 96.240.140.67 (talk) 16:10, 2 March 2021 (UTC)

As a result of the symmetry of the seventh roots of unity, there are four distinct real parts. Your equation has only three solutions.—Anita5192 (talk) 17:30, 2 March 2021 (UTC)

Correction, I meant the primitive roots of unity. The missing fourth zero is x = 1. — Preceding unsigned comment added by 96.240.140.67 (talk) 17:48, 2 March 2021 (UTC)

The only difference between your method and that of the article is that you write the equation of the real part and the article writes the equation of twice the real part, that is the sum of a primitive root and its consjugate. The advantages of the method of the article are: the cubic equation is simpler and it makes clear that its roots are algebraic integers (this is often important in applications). So I see no reason for changing this section of the article. D.Lazard (talk) 18:22, 2 March 2021 (UTC)

Were the references listed twice?

@Anita5192: I noticed that all the links in the "notes" section were references in the article, so it seemed pointless to include them twice. I'm sorry for troubling you if I'm wrong, but it seems wrong to include duplicated information. — Preceding unsigned comment added by MEisSCAMMER (talk • contribs) 23:36, 15 March 2021 (UTC)

The Notes section contains citations of sources from the article. The References section contains mostly different sources. In this case, not all of the sources in the References section are cited, but that does not mean they should be removed. An editor may want to cite them in the future.—Anita5192 (talk) 00:11, 16 March 2021 (UTC)

Okay, my bad then. (Sorry for not signing that other comment -- I don't know what I was thinking.) MEisSCAMMER^(talk)_Hello! 12:15, 16 March 2021 (UTC)

Plot of complex functions

In § Trigonometric expression the plots of the complex functions $z\mapsto z^{3}-1$ and $z\mapsto z^{5}-1$ have been recently changed. It is unclear whether the new version is better. In any case, all these plots are difficult to interpret and of no help to understand the article, which is about equations $z^{n}-1=0,$ not functions $z\mapsto z^{5}-1.$ So MOS:PERTINENCE applies, and I'll remove these images. D.Lazard (talk) 09:00, 7 December 2021 (UTC)

Second root of unity

The article lists -1 as the only primitive second root of unity. To the best of my understanding, -i^2 is also 1, so shouldn't that be a second one?

Wanted to check rather than editing the article in case I'm misunderstanding something. 185.248.67.197 (talk) 19:19, 22 March 2022 (UTC)

$i^{2}=(-i)^{2}=-1.$ So, 1 and −1 are the only square roots of unity, and only −1 is a primitive second root of unity. i and −i are the two primitive fourth roots of unity. It seems that you confuse $-(i^{2})$ and $(-i)^{2}.$ D.Lazard (talk) 21:11, 22 March 2022 (UTC)

Thirds Roots of Unity Figure

In the figure "The 3rd roots of unity" W^1 and W^2 are reversed and need to be swapped. The figure here: https://commons.wikimedia.org/wiki/Category:Roots_of_unity#/media/File:3rd_roots_of_unity_correction.svg is correct. References: G0172 (talk) 16:12, 21 June 2022 (UTC)

There is nothing wrong in the figure, as W^1 and W^2 are both primitive roots of unity, and there is no reason (except common habits) for preferring the one with positive imaginary part. Nevertheless you could have changed the figure yourself. D.Lazard (talk) 17:34, 21 June 2022 (UTC)

References

Reducibility of polynomials of roots of unity

One section says the polynomials are “irreducible”, but I think it needs to be clarified.

If one writes the nth roots of unity in e notation, then the factors x – e^(-2kπi/n) and x – e^(2kπi/n) can be combined to give a polynomial over the real numbers. The sum of the two roots above is real because they are symmetrical about the real axis.

For example: (x^2 + φx + 1)(x^2 + φ^-1x + 1)(x - 1) = x^5 - 1 where φ = (1+ sqrt5)/2 (the golden ratio) Ziconium (talk) 17:04, 16 March 2023 (UTC)

This is the irreducibility over the integers (or equivalently, over the rational numbers) that is considered here. I have clarified this in the article. D.Lazard (talk) 17:47, 16 March 2023 (UTC)

Asking about my latest edit which got reverted

D.Lazard had reverted my latest edit. And whilst I don't mind the revert itself. I'd be glad if the revert reason can be elaborated further than simply WP:ELNO.

Best regards. Jothefiredragon (talk) 09:35, 8 January 2024 (UTC)

In WP:ELNO you will find several reasons for the revert. In WP:Reliable sources, you will find further reasons. On the other hand, you did not provide any reason for including this particular reference (their are thousands of similar courses, and I see no reason for emphasizing this particula one). D.Lazard (talk) 09:56, 8 January 2024 (UTC)

"√1" listed at Redirects for discussion

The redirect √1 has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2025 December 1 § √1 until a consensus is reached. –Deacon Vorbis (carbon • videos) 16:33, 1 December 2025 (UTC)

Reorganizing the lead section, adding sourced info about the link to general nth roots, moving sloppy technical details away from the lead

After requests by D.Lazard and Slawomir Bialy, here I repeat and summarize my motivations for my last modifications:

• the geometric view (about vertices of polygons) is simple enough and important(and related to the picture shown just next the lead section)
• the information about general roots is relevant (as per my sources); the added section in "Elementary properties" about why complex roots of unity can be used to get the polar form of the nth roots of any complex number is a fundamental reason why nth roots of unity are important, which is why the nth roots of unity is introduced in most textbooks (like the ones cited). It was added and sourced (by me); sourced info shouldn't be deleted
• "n-th roots" is more common than just "roots", the article should introduce "n-th roots of unity" directly (and not just in the first subsection), since roots of unity are rarely used without mentioning n
• the setting of complex numbers (first mentioned in the lead) needs to be separated from the setting of more general fields (which should be mentioned only at the end of the lead, and in the definition section)
• the paragraph in the lead about "positive characteristics" and "algebraically closed fields" is much too technical (per MOS:MATH) to figure in the lead section (and it's also very sloppily written and confusing), and should be moved it to the General definition section (as I did). (by contrast, the information i added to the lead section was simple and fundamental)

The sentence ``` If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers.``` in particular is confusing (if it's a general field, why should its elements be complex numbers?). I've kept it temporarily but moved it to the "General definition" section, together with the rest.

The fact that D.Lazard and Slawomir Bialy seem to repeatedly revert the whole of my modifications in bulk with little or no justification despite them being laborious, well-sourced, justified and improving the clarity of the article tends to be interpreted by me as a form of vandalism and a downgrade for the encyclopedia.

Until further notice I reinstate my changes; please do not undo everything in bulk but only the points you disagree with (and explain why). --.NetStacks (talk) 18:23, 3 October 2026 (UTC)

Dear editors. I have restored the last stable version. Please refrain from changing the article until you work it out here. The onus is all three of you to collaborate.

• @.NetStacks: It looks like most of your edits are helpful, but you overwrote some of the existing text without saying why it should no longer be in the article. Note, your edits are not lost, they are still there in the history.
• D.Lazard and Slawomir Bialy, it would be good if you looked at all of .NetStacks's edits and indicated which edits could be restored without objection.
• I suggest starting with the geometric view (about vertices of polygons) is simple enough and important. As far as I can tell, that is correct and useful.

Constant314 (talk) 22:18, 3 October 2026 (UTC)

Administrator comment: Everybody keep WP:3RR in mind. Yue🌙 (talk) 23:19, 3 October 2026 (UTC)

For the record, I was not responsible for the last few reverts. However, I have gone through the disputed changes individually, and I do not support restoring any of that prose remaining. There seems to be a mistaken premise that because material is sourced it should therefore remain (I dispute that a handful of bare URLs to online self-published lecture notes constitutes "well-sourced" material). But WP:ONUS places the burden for inclusion on the editor wishing to include the material (see also WP:BRD, as D.Lazard also indicated). My objections are as follows:

• The proposed opening changes the subject being defined. The article is about a root of unity, a number having a positive integral power equal to one. An nth root of unity is the more specific object obtained after fixing n. Replacing the clean existential definition of the article with "an nth root of unity..." and then immediately switching back to "a root of unity... for some positive integer n" conflates the two notions and produces a redundant and needlessly confused opening. The assertion that "nth roots" are "more common" is unsupported; many sources use "root of unity" both with and without reference to the exponent in different places. Sometimes the specific order is relevant, sometimes it is not; compare with cyclic group: often the order of the group is important and often not, the first sentence is not "a cyclic group of order n is...".
• The material about extracting arbitrary nth roots is substantially off-topic, and at the very least undue in prominence in the article. The article is root of unity, not nth root. The proposed Elementary properties includes a mini-derivation of all nth roots of an arbitrary complex number, with two displayed versions of the same formula and a discussion of how to calculate the remaining roots from one known root. That subject already has an article, Nth root, including sections specifically on complex roots, roots of unity, and nth roots. Reproducing that derivation here is unnecessary duplication and moves the article towards textbook exposition rather than encyclopedic summary.
• That added derivation also includes mathematical errors. It says ${\sqrt[{n}]{r}}$ is "the principal nth root of c", when what is intended is the positive nth root of r.
• The sources do not establish the claimed weight or importance. Indeed, most sources I have that cover roots of unity in detail do not include this as "a fundamental reason nth roots of unity are important" nor "why the nth roots of unity [are] introduced in most textbooks". For example Lang's Algebra studies roots of unity first as the template for abelian extensions, and then applications to Hilbert 90 and Kummer theory. The latter are indeed related to the question of determining all roots, but are not elementary and should be presented in a separate section (if at all), possibly with an introductory and summary-style paragraph pointing to nth roots. Scattered lecture notes or bare urls to page scans don't really establish much WP:WEIGHT.
• The proposed lead is not particularly successful under MOS:MATH. This says that the lead should be accessible to a general reader, introduce the subject informally, and avoid symbols where possible. It does not say that every elementary fact belongs in the lead; it says to avoid overly specific descriptions, and save greater detail for the body of the article. The proposed lead adds a sequence of separate facts about counts of roots, extraction of arbitrary complex roots, etc, but has no narrative cohesion and is a poor summary of the article proper. Just because some fact happens to be elementary does not automatically make it lead-worthy.
• The geometric description is confusing and poorly written. I think something could probably be added to the lead, without the complete demolition of it, but bundling one fixable edit into a mass of poor edits is not a good starting point for fixing it. One reason to initiate a discussion rather than engage in an edit war with multiple editors is to find areas of common ground, and I think this is the most likely place where some accord could be reached.

So, in response to the request to indicate which portions I think can simply be restored: none of the disputed additions can be restored verbatim at this point. That does not preclude proposing individual changes one at a time, for example a particular sentence about the geometry, with exact wording and a rationale. But a large collection of disputed edits should not be treated as presumptively accepted merely because some parts of it are not as bad as others or because some parts of it have (very weak) sources or are elementary.

Finally, I strongly object to describing this good-faith content dispute as "vandalism". Sławomir Biały (talk) 08:04, 4 October 2026 (UTC)

I fully agree with Sławomir Biały's post. Here are some further points

• The restored stable version has multiple issues, and some parts of NetStacks's edits make these worse. The next list item exemplifies this well.
• First paragraph: In the restored version, the first paragraph has several issues that I have fixed in a "undo" of some NetStacks's edit:
  • The third sentence about "de Moivre numbers" was grammatically incorrect. I fixed it — kept by NetStacks, but destroyed by the final revert.
  • The second sentence (context) and the third deserve to be a specific paragrapn. NetStacks's edit removed completely the context.
  • The first sentence requires some thinking to be deciphered by many user. So, I added That is, a root of unity is a number $z$ such that $z^{n}=1$ for some positive integer $n$ — kept with a modification (to which I disagree) by NetStacks, but destroyed by the final revert.
• Complex expression: The trigonomrtric representation of roots of unity is almost never used in texts about roots of unity, and the complex exponential representation is generally used only as a tool for specifying a specific root of unity. So their place at the true beginning of § General definition is WP:UNDUE — NetStacks'edits made this worse.
• Geometric interpretation: This is a very minor aspect of roots of unity, except for understanding some proofs. So, for the lead, the first figure seems suffice, with the caption replaced with The nth roots of unity (here, the 5th roots, in blue) form a regular polyhedron in the complex plane. For the body § Trigonometric expression should be replaced with a section "Geometric representation in the complex plane".

D.Lazard (talk) 11:14, 4 October 2026 (UTC)