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 \(n\geq 1\).


11R09 Polynomials (irreducibility, etc.)
37P15 Dynamical systems over global ground fields
Full Text: DOI arXiv


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