An iterative construction of irreducible polynomials reducible modulo every prime

TL;DR

This paper presents a method to construct polynomials that are globally irreducible but reducible modulo every prime, revealing new phenomena in polynomial iteration and Galois representations over global fields.

Contribution

It introduces criteria for iterates of quadratic polynomials to be irreducible over a global field while reducible modulo all primes, with explicit constructions and examples.

Findings

Constructed infinitely many quadratic polynomials with all iterates irreducible over the field

Provided examples where iterates are irreducible over rationals but reducible modulo primes

Suggested the absence of a rigidity phenomenon in Galois representations from polynomial iteration

Abstract

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 at least 2. We also give an example for each n of a quadratic f with integer coefficients whose iterates are all irreducible over the rationals, 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 P for which a given quadratic f defined over a global field has f^n irreducible modulo P for all n.