A note on Ritt's theorem on decomposition of polynomials

Abstract

It is known (Ritt, 1923; Engstrom, 1941; Lévi, 1942; Dorey and Whaples, 1972) that over fields of characteristic zero, if a polynomial f( x) can be decomposed in two different ways as f= f 1∘ f 2= g 1∘ g 2, then (up to linear transformations) either f 1, f 2, g 1 and g 2are all trigonometric polynomials, or f 1∘ f 2= g 1∘ g 2 is of the form x m ∘ x r · f( x)= x r ·( f( x)) m ∘ x m . The result holds over fields of prime characteristics 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.