Irreducibility of the iterates of a quadratic polynomial over a field

Mohamed Ayad; Donald L. McQuillan

Acta Arithmetica (2000)

  • Volume: 93, Issue: 1, page 87-97
  • ISSN: 0065-1036

Abstract

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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.

How to cite

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Mohamed Ayad, and Donald L. McQuillan. "Irreducibility of the iterates of a quadratic polynomial over a field." Acta Arithmetica 93.1 (2000): 87-97. <http://eudml.org/doc/207402>.

@article{MohamedAyad2000,
abstract = {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.},
author = {Mohamed Ayad, Donald L. McQuillan},
journal = {Acta Arithmetica},
keywords = {irreducibility; quadratic polynomial; stable polynomial},
language = {eng},
number = {1},
pages = {87-97},
title = {Irreducibility of the iterates of a quadratic polynomial over a field},
url = {http://eudml.org/doc/207402},
volume = {93},
year = {2000},
}

TY - JOUR
AU - Mohamed Ayad
AU - Donald L. McQuillan
TI - Irreducibility of the iterates of a quadratic polynomial over a field
JO - Acta Arithmetica
PY - 2000
VL - 93
IS - 1
SP - 87
EP - 97
AB - 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.
LA - eng
KW - irreducibility; quadratic polynomial; stable polynomial
UR - http://eudml.org/doc/207402
ER -

References

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  1. [1] E. Artin and J. Tate, Class Field Theory, Benjamin, New York, 1968. Zbl0176.33504
  2. [2] M. Ayad, Théorie de Galois, 122 exercices corrigés, niveau I, Ellipses, Paris, 1997. 
  3. [3] Z. I. Borevitch et I. R. Chafarevitch, Théorie des nombres, Gauthier-Villars, Paris, 1967. 
  4. [4] Y. Hellegouarch, Loi de réciprocité, critère de primalité dans q [ t ] , C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 291-296. Zbl0608.12022
  5. [5] P. J. McCarthy, Algebraic Extensions of Fields, Blaisdell, Waltham, 1966. Zbl0143.05802
  6. [6] R. W. K. Odoni, On the prime divisors of the sequence w n + 1 = 1 + w ₁ . . . w n , J. London Math. Soc. 32 (1985), 1-11. Zbl0574.10020
  7. [7] R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. 51 (1985), 385-414. Zbl0622.12011
  8. [8] O. Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc. 36 (1934), 243-274. Zbl60.0111.04
  9. [9] N. G. Tschebotaröw, Grundzüge der Galois'schen theorie (translated from Russian by H. Schwerdtfeger), Noordhoff, Groningen, 1950. Zbl0037.14602

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