Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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COMPOSITION OF BINOMIAL POLYNOMIAL

  • Choi, Eun-Mi (DEPARTMENT OF MATHEMATICS HAN NAM UNIVERSITY)
  • Published : 2007.04.30
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Abstract

For an irreducible binomial polynomial $f(x)=x^n-b{\in}K[x]$ with a field K, we ask when does the mth iteration $f_m$ is irreducible but $m+1th\;f_{m+1}$ is reducible over K. Let S(n, m) be the set of b's such that $f_m$ is irreducible but $f_{m+1}$ is reducible over K. We investigate the set S(n, m) by taking K as the rational number field.

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References

  1. F. Beukers, The Diophantine equation $Ax^P$ + $By^q$ = $Cz^T$, Duke Math. J. 91 (1998), 61-88 https://doi.org/10.1215/S0012-7094-98-09105-0
  2. N. Bruin, The Diophantine equations $x^2 {\pm} y^4 = {\pm}z^6$ and $x^2 + y^8 = z^3$, Compositio Math. 118 (1999), 305-321 https://doi.org/10.1023/A:1001529706709
  3. N. Bruin, On powers as sums of two cubes, in Algorithmic Number theory, Prod. 4th International Symp. Lecture Notes in Computer Science 1838, Springer, New York, (2000), 169-184
  4. L. Danielson and B. Fein, On the irreducibility of the iterates of $x^n$ - b, Proc. Amer. Math. Soc. 130 (2001), 1589-1597 https://doi.org/10.1090/S0002-9939-01-06258-X
  5. H. Darmon, The equation $x^4-y^4=z^p$, C.R.Math. Rep. Acad. Sci. Canada 15 (1993), 286-290
  6. H. Darmon, The equation $x^n+y^n=z^2\;and\;x^n+y^n=z^3$, Int. Math. Res. Notices 10 (1993), 236-274
  7. 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 https://doi.org/10.1112/blms/27.6.513
  8. H. Darmon and L. Merel, Widning quotients and some variants of Fermat's Last Theorem, J. Reine Angew. Math. 490 (1997), 81-100
  9. B. Fein and M. Schacker, Properties of iterates and composites of polynomials, J. London Math. Soc. 54 (1996), 489-497 https://doi.org/10.1112/jlms/54.3.489
  10. A. Kraus, Sur l'equation $a^3+b^3=c^p$, Experiment Math. 7 (1998), 1-13 https://doi.org/10.1080/10586458.1998.10504355
  11. A. Kraus, On the equation $x^p+y^p=z^r$, A survey, Ramanujan 3 (1999), 315-333 https://doi.org/10.1023/A:1009835521324
  12. R. W. K. Odoni, The Galois theory of iterates and composites of polynomials, Proc. London Math. Soc. 51 (1985), 385-414 https://doi.org/10.1112/plms/s3-51.3.385
  13. R. W. K. Odoni, Realising wreath products of cyclic groups as Galois groups, Mathematika 35 (1988), 101-113 https://doi.org/10.1112/S002557930000632X
  14. B. Poonen, Some Diophantine equations of the form $x^n+y^n=z^m$, Acta Arith. LXXXVI 3 (1998), 193-205
  15. M. Stoll, Galois groups over Q of some iterated polynomials, Arch. Math. 59 (1992), 239-244 https://doi.org/10.1007/BF01197321