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Stability of polnomials. (Stabilité des polynômes.) (French) Zbl 1088.11078
Let \(E\) be any field. An irreducible polynomial \(f\in E[X]\) is called stable over \(E\) if all its iterates \(f\circ f,f\circ f \circ f,\dots\) are irreducibles in \(E[X]\). The aim of this paper is to investigate the stability of certain irreducible polynomials over algebraic number fields.
More precisely, let \(K\) be an algebraic number field of degree \(n\), and let \(\{w_1,\ldots ,w_n\}\) be an integral basis of \(K\). Let \(\{u_1,\ldots ,u_n\}\) be algebraically independent variables over \(K\), \(L = \mathbb{Q} (u_1,\ldots u_n)\), \(\xi=u_1w_1 + \ldots + u_nw_n\), and let \(F(X)\) be the minimal polynomial of \(\xi\) over \(L\). The author shows that under certain arithmetical conditions on \(K\), there exist infinitely many integral elements \(\alpha \in K\) such that the minimal polynomial of \(\alpha\) over \(\mathbb{Q}\) is stable over \(\mathbb{Q}\), and the polynomial \(F\in L[X]\) is stable over \(L\).

11R09 Polynomials (irreducibility, etc.)
11T06 Polynomials over finite fields
12E10 Special polynomials in general fields
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