an:00950114
Zbl 0865.12003
Fein, Burton; Schacher, Murray
Properties of iterates and composites of polynomials
EN
J. Lond. Math. Soc., II. Ser. 54, No. 3, 489-497 (1996).
00037896
1996
j
12E05
composition; irreducibility; polynomials; iterates; Hilbertian field
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.
B.Fein (Corvallis)