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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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