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On the irreducibility of the iterates of \(x^{n}-b\). (English) Zbl 1007.12001
Suppose that \(f(x) = x^n - b\) is a polynomial irreducible over a field \(K\). The authors investigate under what circumstances all iterates of \(f(x)\) are irreducible over \(K\). This happens, for example, when \(K = {\mathbb Q}\), \(b \in {\mathbb Z}\), when \(K = {\mathbb Q}(t)\) (rational function field) and \(b \in {\mathbb Z}[t]\) (polynomial ring), when \(K = F(t)\) and \(b \in F[t]\), for \(F\) an algebraically closed field, and when \(K = F(t)\), \(b\) in \(F(t)\) but not \(F\), \(n \geq 3\) and \(F\) a field of characteristic 0. The existence of a reducible iterate is tied to the existence of a primitive solution to a Diophantine equation of the form \(x^p + y^p = z^r\) for a suitable prime \(p\).

MSC:
12E05 Polynomials in general fields (irreducibility, etc.)
11D41 Higher degree equations; Fermat’s equation
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