A note on Ritt's theorem on decomposition of polynomials

Abstract

It is known [J. F. Ritt, Trans. Am. Math. Soc. 23, 51-66 (1922; JFM 48.0079.01), H. T. Engstrom, Am. J. Math. 63, 249–255 (1941; Zbl 0025.10403), H.Levi, ibid. 64, 389–400 (1942; Zbl 0063.03512), F. Dorey and G. Whaples, J. Algebra 28, 88-101 (1974; Zbl 0286.12102)] that over fields of characteristic zero, if a polynomial f(x) can be decomposed into two different ways as f = f1 o f2 = g1 o g2, then (up to linear transformations) either f1, f2, g1 and g2 are all trigonometric polynomials, or f1of2 = g1 o g2 is of the form xm o xr · f(x) = xr · (f(x))m o xm. The result holds over fields of prime characteristic when the involved field extensions are separable and there are no wildly ramified primes. In this note we give an example of a whole family of polynomials with degrees non divisible by the characteristic of the field having more than one decomposition.