# Talk:Dickson polynomial

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## Schur conjecture reference

Howdy, I read through the Schur conjecture reference, and it only mentions the Chebyshev polynomials. Are these really just the same polynomials? If so, it seems weird to have separate articles, and if not it seems the reference doesn't support the claim. The material in this article seems interesting, the article on Chebyshev polynomials is already long, and there is a book with the title Dickson Polynomials, so it would be good to keep the articles split, but I'm not sure the relationship is totally clear yet.

More technical detail: so the Schur thingy only requires that one look at infinitely many primes, so it should be no issue to divide by 2, and I think we can assume alpha is a square mod infinitely many of the primes, but I am not sure about whether we can over the rationals. So I am not sure they are the same, just that Chebyshev polynomials are roughly speaking a subset of the Dickson polynomials. Why does the article only talk about Chebyshev polynomials then? Do they prove a stronger result than the one claimed in this wikipedia article? JackSchmidt (talk) 19:35, 8 February 2008 (UTC)

In response to JackSchmidt's question: the reason Fried's article only talks about Chebyshev polynomials is that this is one of several errors in that article. The theorem that Fried stated is false, and to salvage it one must use Dickson polynomials. For instance, as noted on pp.259-260 of Turnwald's paper "On a problem concerning permutation polynomials" (Transactions of the American Mathematical Society 302, number 1, 1987), Fried uses three pairwise inequivalent definitions of Chebyshev polynomials. A correct statement and proof of the so-called "Schur conjecture" can be found in the cited book on Dickson polynomials, among other places.

More generally, the whole point of Dickson polynomials is that you can take a Chebyshev polynomial over a given field, compose it on both sides with linear polynomials defined over an extension field, and wind up with a polynomial over the original field. For instance, the degree-3 Chebyshev polynomial is T_3(x)=4x^3-3x, and if you compose on both sides with ix then you get i * (4(ix)^3-3(ix)) = 4x^3+3x. The resulting polynomial 4x^3+3x has rational coefficients, but cannot be written as the composition of 4x^3-3x with linears in Q[x]. It is thus a "twist" of 4x^3-3x, and Dickson polynomials exist to capture such twists: indeed, 4x^3+3x=D_3(2x,-1)/2 is the composition of D_3(x,-1) with linears in Q[x]. ==Mike Zieve

I suggest refreshing the section :Links to other polynomials as follows (demonstration available), is there any objection?

${\displaystyle D_{n}(2xa,a^{2})=2a^{n}T_{n}(x)\,}$
${\displaystyle E_{n}(2xa,a^{2})=a^{n}U_{n}(x).\,}$
${\displaystyle B_{n+1}(x)B_{n+j}(x)-B_{n+j+1}(x)B_{n}(x)=(3x^{2}+4)D_{n+1}(x,{\frac {1}{4}})\,}$
${\displaystyle B_{n}\left(x\right)=D_{n}(2x,{\frac {1}{4}})+4D_{n-1}(2x,{\frac {1}{4}})\,}$

Etaittunpe (talk) 01:39, 15 February 2009 (UTC)

Nonsense. As I pointed out, those Dickson polynomials are essentially Chebychev polynominals, with the relation:
${\displaystyle D_{n}(x,{\frac {1}{4}})=2^{1-n}T_{n}(x)}$
Arthur Rubin (talk) 10:33, 15 February 2009 (UTC)

## Schur conjecture name

The name "Schur conjecture" is inappropriate. Schur did not make this conjecture, and in fact he took pains to avoid making any such conjecture. Specifically, in his 1923 paper "Ueber den Zusammenhang zwischen einem Problem der Zahlentheorie und einem Satz ueber algebraische Funktionen", Schur called a positive integer n a "Dickson number" if every degree-n polynomial in Z[x] which induces a permutation of Z/pZ for infinitely many primes p is a composition of Dickson polynomials and linears. He proved that prime numbers are Dickson numbers. He also asserted that he would prove in a subsequent paper that a positive integer n is a Dickson number if it has the property that for each (positive) composite divisor d of n, every permutation group on d symbols which contains a d-cycle is either doubly transitive or imprimitive. Unfortunately, Schur never published this proof, nor did he mention Dickson numbers again in his papers. Further, Schur made no conjecture about which positive integers n have the stated group-theoretic property. He did, however, prove some years later that *every* positive integer n has this property: for every composite d, every permutation group on d symbols which contains a d-cycle is either doubly transitive or imprimitive. This group-theoretic result is now known as "Schur's theorem", and it plays a crucial role in all known proofs of the so-called "Schur conjecture".

The term "Schur conjecture" entered the literature in two Mathematical Reviews written by Don Lewis in 1966. I have spoken with Don about this, and he says that he had no special knowledge of Schur making this conjecture, but instead that he (Don) must have misremembered something he was told by Davenport. A few years later, Don's student Mike Fried wrote a paper on the so-called "Schur conjecture", and in the years that followed this terminology became commonplace. However, it is not accurate: Schur did not make any such conjecture. (Someone new to the topic might be confused by the fact that Mike Fried cites a specific page of Schur's paper in his article; however, there is no conjecture to be found on that or any other page of Schur's paper.) ==Mike Zieve

## Permutation polynomials

"Dn is a permutation polynomial for the field with q elements if and only if n is coprime to q²−1".

Isn't there the extra condition that q must be prime ? — Preceding unsigned comment added by Fabrice.Neyret (talkcontribs) 06:18, 4 April 2017 (UTC)

No, finite fields exist whenever q is a power of a prime. When q is a prime you get the simplest examples of the finite fields, which are the integers modulo that prime. The other finite fields can be viewed as field extensions of these basic examples.--Bill Cherowitzo (talk) 15:53, 16 April 2017 (UTC)