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On the irreducibility of the iterates of $x^{n}-b$

Authors: Lynda Danielson and Burton Fein
Journal: Proc. Amer. Math. Soc. 130 (2002), 1589-1596
MSC (2000): Primary 12E05; Secondary 11D41
Published electronically: October 17, 2001
MathSciNet review: 1887002
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Abstract: Let $K$ be a field and suppose that $f(x)=x^{n}-b$ is irreducible in $K[x]$. We discuss the following question: under what conditions are all iterates of $f$ irreducible over $K$?

References [Enhancements On Off] (What's this?)

  • [B] Nils Bruin, On powers as sums of two cubes, in Algorithmic Number Theory, Proceedings 4th International Symposium, ANTS-IV (W. Bosma, ed.), Lecture Notes in Computer Science 1838, Springer, New York, 2000, pp. 169-184.
  • [DG] H. Darmon and A. Granville, On the equations $z^{m}=F(x,y)$ and $Ax^{p}+By^{q}=Cz^{r}$, Bull. London Math. Soc. 27 (1995), 513-543. MR 96e:11042
  • [FS] B. Fein and M. Schacher, Properties of iterates and composites of polynomials, J. London Math. Soc. 54 (1996), 489-497. MR 97h:12007
  • [K] A. Kraus, On the equation $x^{p}+y^{q}=z^{r}$, A survey, Ramanujan J. 3 (1999), 315-333. MR 2001f:11046
  • [L] S. Lang, Algebra, third edition, Addison-Wesley, Reading, 1993. MR 33:5416 (review of first edition); MR 86j:00003 (review of second edition)
  • [O] R. W. K. Odoni, Realizing wreath products of cyclic groups as Galois groups, Mathematika 35 (1988), 101-113. MR 90a:12013
  • [S] M. Stoll, Galois groups over $\mathbb{Q}$ of some iterated polynomials, Arch. Math. 59 (1992), 239-244. MR 93h:12004

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Additional Information

Lynda Danielson
Affiliation: Department of Mathematics, Albertson College of Idaho, Caldwell, Idaho 83605

Burton Fein
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Keywords: Iterated polynomial, irreducible, generalized Fermat equation, $abc$-conjecture
Received by editor(s): April 13, 2000
Received by editor(s) in revised form: December 20, 2000
Published electronically: October 17, 2001
Additional Notes: The second author is grateful for support under NSA Grant MDA904-97-1-0040
Communicated by: Lance W. Small
Article copyright: © Copyright 2001 American Mathematical Society

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