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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 \(\mathbb Q\).

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