A note on Ritt's theorem on decomposition of polynomials
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1990
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Elsevier Science B.V. (North-Holland)
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F. Dorey and G. Whaples, Prime and composite polynomials,J. Algebra 28 (1972) 88-101.
H.T. EngstrBm, Polynomial substitutions, Amer. J. Math.63 (1941) 249-255.
M. Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970) 41-55.
H. Levi, Composite polynomials with coefficients in an arbitrary field of characteristic zero, Amer.J. Math. 64 (1942) 389-400.
J.F. Ritt, Prime and composite polynomials, Trans. Amer.Math. Sot. 23 (1922) 51-66.
A. Schinzel, Selected Topics in Polynomials (Section 4,Ritt’s First Theorem; Section 5, Ritt’s Second Theorem) (University of Michigan Press, Ann Arbor, 1982) 12-39.
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.