Talk:Cyclotomic polynomial

Created page

The two "citation needed" things are definitely true but the only references I have are to out-of-print books.

Clean-up requested, needs citation for record coefficients.

I just added a request for clean-up because of the monstrous paragraph at the end of the Properties section is a rambling mess. Additionally, it contains a reference to a very recent record-height polynomial. Computing such large polynomials requires dozens of hours of computation time and many gigabytes of memory, so this is probably original research. 142.58.12.53 (talk) 20:31, 24 January 2011 (UTC)[reply]

Yes, it IS original research, and I added the source now in two weblinks. Maybe there is a mistake in formatizing, because the URLs begin Www.... (talk) 22:19, 25 January 2011 (UTC)[reply]

Original research is not allowed on Wikipedia. See WP:NOR and WP:IRS. See also WP:COI.—Emil J. 11:27, 26 January 2011 (UTC)[reply]

Misapplication of Möbius inversion?

I'm concerned that this assertion [in `Fundamental Tools']

> The Möbius inversion formula yields the equivalent formulation: > > \prod_{d\mid n}(x^d-1)^{\mu(n/d)} = \Phi_n(x) >

is QUITE FALSE?! It seems to imply, for example, that \Phi_n(1) = 0 for any n, whereas the first assertion on the page

> If n is a prime number then > > ~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.

implies that \Phi_p(1) = p if n = p is prime.

The mistake seems to originate in a misapplication of the M\"obius inversion formula, which would require that

> \Phi_{mn}(x) = \Phi_m(x)\Phi_n(x)

for m,n coprime; in other words, that \Phi_n is a `multiplicative' arithmetic function. This fails, for example, if m = 2 and n is an odd prime.

This error seems to be recur in the article on M\"obius inversion, where the requirement that the function be multiplicative is not mentioned. This property is sometimes taken as part of the definition of an `arithmetic function', which may be where this error crept in.

       ++++++++++++++++++++++++

I hope I am mistaken. I am inexperienced in the ways of Wikipedia, and suspect this entry requires the attention of an expert in number theory = which I am not. I hope the readers, if any, of this message will bring it to the attention of some such person.

Thanks for your attention! — Preceding unsigned comment added by Drwonmug (talk • contribs) 18:01, 28 January 2013 (UTC)[reply]

Thank you for bringing our attention on this paragraph. The results concerning cyclotomic polynomials are correct, but the explanation was silly. I have corrected this by removing the reference to Euler totient function which were misleading and linking to the relevant paragraph of Möbius inversion formula. D.Lazard (talk) 08:38, 29 January 2013 (UTC)[reply]

Möbius inversion does not require the two functions to be multiplicative, or to have any other special property. The formula

\prod _{d\mid n}(x^{d}-1)^{\mu (n/d)}=\Phi _{n}(x)

is an equality of two rational functions (i.e., elements of

\mathbb {Q} (x)

), equivalently it can be stated in the form that the polynomial on the right-hand side is the ratio of two polynomials on the left-hand side. If you want to evaluate such an equality at an arbitrary point which may be a root of the denominator, you cannot just plug it in and expect it to work. For example,

{\frac {x^{2}-1}{x-1}}=x+1

is correct as polynomial division, but if you substitute 1 for x, you get 0/0.—Emil J. 15:24, 29 January 2013 (UTC)[reply]

Prime Cyclotomic numbers −> Values

The section "Prime Cyclotomic numbers", introduced in April 2014 by an IP user was essentially original research, consisting essentially in reproducing and commenting some OEIS tables. As these tables were relevant to the subject, I did not removed the section, and I have only tagged it as unreferenced. A reason for not removing the section was that there were no other section devoted to the study of the values of cyclotomic polynomials.

Recently, when trying to wikify Unique prime (an article probably introduced by the same IP user), I found interesting properties of the values of the cyclotomic polynomials, which were stated (partially incorrectly) without explanation nor reference. As I have no access to the references of Unique prime, I have provided a proof of these properties. I ignore if they were published elsewhere, but, as the proof is rather elementary, this is highly probable. As these properties are relevant for a section of the values of cyclotomic polynomials, I have introduced them as a new section § Values, which contains also links to the OEIS tables of the preceding section.

References for these results would be welcome. D.Lazard (talk) 15:47, 21 October 2016 (UTC)[reply]

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Example

The example sections gives two formulas, one for when n is prime, and the other for n is 2p, where p is an odd prime. But it doesn't give a formula for when neither condition applies. Could somebody add this? MiguelMunoz (talk) 06:46, 17 September 2023 (UTC)[reply]

This does not belong to section § Examples. Two general formulas are given in § Fundamental tools. D.Lazard (talk) 11:42, 17 September 2023 (UTC)[reply]