×

zbMATH — the first resource for mathematics

An iterative construction of irreducible polynomials reducible modulo every prime. (English) Zbl 1302.11086
Summary: We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field \(F\) but reducible modulo every prime of \(F\). The method consists of finding quadratic \(f\in F[x]\) whose iterates have the desired property, and it depends on new criteria ensuring all iterates of \(f\) are irreducible. In particular when \(F\) is a number field in which the ideal (2) is not a square, we construct infinitely many families of quadratic \(f\) such that every iterate \(f^{n}\) is irreducible over \(F\), but \(f^{n}\) is reducible modulo all primes of \(F\) for \(n\geq 2\). We also give an example for each \(n\geq 2\) of a quadratic \(f\in {\mathbb Z}[x]\) whose iterates are all irreducible over \(\mathbb Q\), whose \((n-1)\)st iterate is irreducible modulo some primes, and whose nth iterate is reducible modulo all primes. From the perspective of Galois theory, this suggests that a well-known rigidity phenomenon for linear Galois representations does not exist for Galois representations obtained by polynomial iteration. Finally, we study the number of primes \(\mathfrak p\) for which a given quadratic \(f\) defined over a global field has \(f^{n}\) irreducible modulo \(\mathfrak p\) for all \(n\geq 1\).

MSC:
11R09 Polynomials (irreducibility, etc.)
37P15 Dynamical systems over global ground fields
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Ahmadi, Omran; Luca, Florian; Ostafe, Alina; Shparlinski, Igor, On stable quadratic polynomials, Glasg. math. J., 54, 2, 359-369, (2012) · Zbl 1241.11027
[2] Ali, Nidal, Stabilité des polynômes, Acta arith., 119, 1, 53-63, (2005) · Zbl 1088.11078
[3] Ayad, Mohamed; McQuillan, Donald L., Irreducibility of the iterates of a quadratic polynomial over a field, Acta arith., 93, 1, 87-97, (2000) · Zbl 0945.11020
[4] Ayad, Mohamed; McQuillan, Donald L.; Ayad, Mohamed; McQuillan, Donald L., Corrections to: “irreducibility of the iterates of a quadratic polynomial over a field”, Acta arith., Acta arith., 99, 1, 97-97, (2001) · Zbl 0945.11020
[5] Boston, Nigel; Jones, Rafe, Settled polynomials over finite fields, Proc. amer. math. soc., 140, 6, 1849-1863, (2012) · Zbl 1243.11115
[6] Brandl, Rolf, Integer polynomials that are reducible modulo all primes, Amer. math. monthly, 93, 4, 286-288, (1986) · Zbl 0603.12002
[7] Danielson, Lynda; Fein, Burton, On the irreducibility of the iterates of \(x^n - b\), Proc. amer. math. soc., 130, 6, 1589-1596, (2002), (electronic) · Zbl 1007.12001
[8] Fein, Burton; Schacher, Murray, Properties of iterates and composites of polynomials, J. lond. math. soc. (2), 54, 3, 489-497, (1996) · Zbl 0865.12003
[9] Flajolet, Philippe; Odlyzko, Andrew M., Random mapping statistics, (), 329-354 · Zbl 0747.05006
[10] Guralnick, Robert; Schacher, Murray M.; Sonn, Jack, Irreducible polynomials which are locally reducible everywhere, Proc. amer. math. soc., 133, 11, 3171-3177, (2005), (electronic) · Zbl 1134.11040
[11] Jones, Rafe, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. lond. math. soc. (2), 78, 2, 523-544, (2008) · Zbl 1193.37144
[12] Jones, Rafe; Rouse, Jeremy, Galois theory of iterated endomorphisms, Proc. lond. math. soc. (3), 100, 3, 763-794, (2010), Appendix A by Jeffrey D. Achter · Zbl 1244.11057
[13] Neukirch, Jürgen, Algebraic number theory, Grundlehren math. wiss., vol. 322, (1999), Springer-Verlag Berlin, translated from the 1992 German original and with a note by Norbert Schappacher, with a foreword by G. Harder · Zbl 0956.11021
[14] Odoni, R.W.K., The Galois theory of iterates and composites of polynomials, Proc. lond. math. soc. (3), 51, 3, 385-414, (1985) · Zbl 0622.12011
[15] Odoni, R.W.K., On the prime divisors of the sequence \(w_{n + 1} = 1 + w_1 \cdots w_n\), J. lond. math. soc. (2), 32, 1, 1-11, (1985) · Zbl 0574.10020
[16] Odoni, R.W.K., Realising wreath products of cyclic groups as Galois groups, Mathematika, 35, 1, 101-113, (1988) · Zbl 0662.12010
[17] Ostafe, Alina; Shparlinski, Igor E., On the length of critical orbits of stable quadratic polynomials, Proc. amer. math. soc., 138, 8, 2653-2656, (2010) · Zbl 1268.11155
[18] Rosen, Michael, Number theory in function fields, Grad. texts in math., vol. 210, (2002), Springer-Verlag New York · Zbl 1043.11079
[19] Silverman, Joseph H., Variation of periods modulo p in arithmetic dynamics, New York J. math., 14, 601-616, (2008) · Zbl 1153.11028
[20] Stoll, Michael, Galois groups over Q of some iterated polynomials, Arch. math. (basel), 59, 3, 239-244, (1992) · Zbl 0758.11045
[21] Vasiu, Adrian, Surjectivity criteria for p-adic representations. I, Manuscripta math., 112, 3, 325-355, (2003) · Zbl 1117.11064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.