An iterative construction of irreducible polynomials reducible modulo every prime

JOURNAL OF ALGEBRA

卷:369

Article

Jones, Rafe

关键词: Algebraic number theory;    Irreducibility of polynomials;    Polynomial iteration;   

DOI  :  10.1016/j.jalgebra.2012.05.020

来源: Elsevier

【 摘 要 】

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 is an element of 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 >= 2. We also give an example for each n >= 2 of a quadratic f is an element of Z[x] whose iterates are all irreducible over 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 p for which a given quadratic f defined over a global field has f(n) irreducible modulo p for all n >= 1. (c) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free