Irreducibility of the iterates of a quadratic polynomial over a field

• Autorzy: Mohamed Ayad , Donald L. McQuillan
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11C08: Polynomials
• 12E05: Polynomials (irreducibility, etc.)
• 11T06: Polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2000
• Tom: 93
• Numer: 1
• Strony: 87-97

Daty

wydano

2000

otrzymano

1999-06-11

poprawiono

1999-09-09

Twórcy

autor

Mohamed Ayad

• Université du Littoral Cote d'Opale, 50, Rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France

autor

Donald L. McQuillan

• Department of Mathematics, University College Dublin, Belfield 4, Dublin, Ireland

Bibliografia

• [1] E. Artin and J. Tate, Class Field Theory, Benjamin, New York, 1968.
• [2] M. Ayad, Théorie de Galois, 122 exercices corrigés, niveau I, Ellipses, Paris, 1997.
• [3] Z. I. Borevitch et I. R. Chafarevitch, Théorie des nombres, Gauthier-Villars, Paris, 1967.
• [4] Y. Hellegouarch, Loi de réciprocité, critère de primalité dans $𝔽_q[t]$, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 291-296.
• [5] P. J. McCarthy, Algebraic Extensions of Fields, Blaisdell, Waltham, 1966.
• [6] R. W. K. Odoni, On the prime divisors of the sequence $w_n + 1 = 1 + w₁...w_n$, J. London Math. Soc. 32 (1985), 1-11.
• [7] R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. 51 (1985), 385-414.
• [8] O. Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc. 36 (1934), 243-274.
• [9] N. G. Tschebotaröw, Grundzüge der Galois'schen theorie (translated from Russian by H. Schwerdtfeger), Noordhoff, Groningen, 1950.