Irreducibility of the iterates of a quadratic polynomial over a field

Mohamed Ayad; Donald L. McQuillan

Acta Arithmetica, Tome 92 (2000), p. 87-97 / Harvested from The Polish Digital Mathematics Library

Résumé

1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define $f_{r + 1} (X) = f (f_{r} (X))$ for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if $f_{r} (X)$ is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof. In the sequel we shall use elementary methods for proving the stability of quadratic polynomials over number fields; especially the rational field, and over finite fields of characteristic p ≥ 3.

Publié le : 2000-01-01

Zbl 0945.11020

EUDML-ID : urn:eudml:doc:207402

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     author = {Mohamed Ayad and Donald L. McQuillan},
     title = {Irreducibility of the iterates of a quadratic polynomial over a field},
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     volume = {92},
     year = {2000},
     pages = {87-97},
     zbl = {0945.11020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav93i1p87bwm}
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Mohamed Ayad; Donald L. McQuillan. Irreducibility of the iterates of a quadratic polynomial over a field. Acta Arithmetica, Tome 92 (2000) pp. 87-97. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav93i1p87bwm/

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