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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
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