Fein, Burton; Schacher, Murray Properties of iterates and composites of polynomials. (English) Zbl 0865.12003 J. Lond. Math. Soc., II. Ser. 54, No. 3, 489-497 (1996). Suppose \({\mathcal P}\) is a property of polynomials and \(r\) is an arbitrary natural number. This paper is concerned with the following question: does there exist a field \(K\) and a polynomial \(f(x)\in K[ x]\) such that the first \(r\) iterates of \(f(x)\) have property \({\mathcal P}\) but the next iterate does not? (The iterates of \(f(x)\) are defined by \(f_1 (x)= f(x)\) and \(f_{k+1} (x)= f(f_k (x))\) for \(k\geq 1\).) The existence of such examples is proven for several of the most frequently considered properties of polynomials: (a) irreducibility, (b) separability, (c) splitting completely, and (d) solvability by radicals. In these examples, \(K\) may be taken to be Hilbertian. The question of whether such examples exist over a prescribed Hilbertian field (e.g. \(\mathbb{Q}\)) is left unresolved. Reviewer: B.Fein (Corvallis) Cited in 1 ReviewCited in 11 Documents MSC: 12E05 Polynomials in general fields (irreducibility, etc.) Keywords:composition; irreducibility; polynomials; iterates; Hilbertian field PDFBibTeX XMLCite \textit{B. Fein} and \textit{M. Schacher}, J. Lond. Math. Soc., II. Ser. 54, No. 3, 489--497 (1996; Zbl 0865.12003) Full Text: DOI