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Irreducibility of the iterates of a quadratic polynomial over a field. (English) Zbl 0945.11020
Acta Arith. 93, No. 1, 87-97 (2000); correction ibid. 99, No. 1, 97 (2001).
R. W. K. Odoni [Proc. Lond. Math. Soc., III. Ser. 51, 385-414 (1985; Zbl 0622.12011)] defined a polynomial $$f(x)$$ with coefficients in a field $$K$$ to be stable over $$K$$ if every polynomial in the sequence $$f(x), f(f(x)), f(f(f(x))), \dots$$ is irreducible over $$K$$. For instance, the polynomial $$f(x)=x^2-x+1$$ is stable over $$\mathbb Q$$. The proof, however, was quite difficult [R. W. K. Odoni, J. Lond. Math. Soc., II. Ser. 32, 1-11 (1985; Zbl 0574.10020)].
The authors give some sufficient conditions for the quadratic polynomial $$x^2-ax+b$$, where $$a, b \in \mathbb Z$$, with discriminant $$d=a^2-4b$$, to be stable over $$\mathbb Q$$. In Theorem 2 they prove that if $$d-1$$ is divisible by 4, then the polynomial $$x^2-ax+b$$ is stable over $$\mathbb Q$$. This is also the case if $$d$$ is divisible by 4, but is not divisible by 16 (Theorem 3). The proofs are elementary. They also obtain some sufficient conditions for a quadratic polynomial to be stable over $$K$$, where $$K$$ is a finite field or an arbitrary number field. The proofs in these cases are more complicated. In particular, the latter results include two new proofs of stability of the polynomial $$x^2-x+1$$ over $$\mathbb Q$$.

##### MSC:
 11R09 Polynomials (irreducibility, etc.) 11T06 Polynomials over finite fields 12E05 Polynomials in general fields (irreducibility, etc.)