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  • $\begingroup$ Thanks, Keith! I knew that Chebyshev polynomials had this property, but had no idea how special it was. $\endgroup$ Commented May 21, 2010 at 23:30
  • $\begingroup$ Huh, I didn't know that theorem of Ritt's. Just to emphasize the naturality of this point of view: one day during graduate school this question came up (somewhat randomly in discussion) and an officemate and I took one afternoon and proved exactly that. (Okay, not quite: we didn't ask the polynomials to be monic and also just asked for polynomials over the rings Z, Q, R, rather than an arbitrary field of Char 0. But the result is essentially the same. So our proof is considerably shorter [~4 pages].) $\endgroup$ Commented May 22, 2010 at 0:36
  • $\begingroup$ Willie, the article I cited from Kvant Selecta explains the result in just a few pages. $\endgroup$ Commented May 22, 2010 at 16:19
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    $\begingroup$ It's maybe worth pointing out that whilst Ritt's original proofs rely on the topology of Riemann surfaces, it can be done purely algebraically. For references, see eg Clauwens, Commuting polynomials and $\lambda$-ring structures on Z[x] , J. Pure Appl. Algebra 95 (1994). This paper gives the following nice consequence: there are exactly two $\lambda$-ring structures on Z[x], one arising from powers and the other from the Chebyshev polynomials. $\endgroup$ Commented May 22, 2010 at 16:59
  • $\begingroup$ dke: The Kvant article I mention is also a simple treatment of the commuting family of polynomials problem. I gave the Ritt citation as a matter of historical precedence, not because I thought it was the ideal argument. $\endgroup$ Commented May 22, 2010 at 17:49