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## Acta Arithmetica

2000 | 93 | 1 | 87-97
Tytuł artykułu

### Irreducibility of the iterates of a quadratic polynomial over a field

Autorzy
Treść / Zawartość
Warianty tytułu
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
87-97
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-06-11
poprawiono
1999-09-09
Twórcy
autor
• Université du Littoral Cote d'Opale, 50, Rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France
autor
• 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.
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