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We give a method of constructing polynomials of arbitrarily large degreeirreducible over a global field F but reducible modulo every prime of F. Themethod consists of finding quadratic f in F[x] whose iterates have the desiredproperty, and it depends on new criteria ensuring all iterates of f areirreducible. In particular when F is a number field in which the ideal (2) isnot a square, we construct infinitely many families of quadratic f such thatevery iterate f^n is irreducible over F, but f^n is reducible modulo all primesof F for n at least 2. We also give an example for each n of a quadratic f withinteger coefficients whose iterates are all irreducible over the rationals,whose (n-1)st iterate is irreducible modulo some primes, and whose nth iterateis reducible modulo all primes. From the perspective of Galois theory, thissuggests that a well-known rigidity phenomenon for linear Galoisrepresentations does not exist for Galois representations obtained bypolynomial iteration. Finally, we study the number of primes P for which agiven quadratic f defined over a global field has f^n irreducible modulo P forall n.

An iterative construction of irreducible polynomials reducible modulo every prime