An iterative construction of irreducible polynomials reducible modulo every prime
An iterative construction of irreducible polynomials reducible modulo every prime
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不可约多项式的迭代构造可约模每个素数
DOI：
10.1016/j.jalgebra.2012.05.020
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Rafe Jones
中科院分区：
文献类型：
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作者：
Rafe Jones
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∈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 fnis irreducible over F, but fnis reducible modulo all primes of F for n⩾2. We also give an example for each n⩾2 of a quadratic f∈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 fnirreducible modulo p for all n⩾1.
