Given a field K and n > 1, we say that a polynomial f ∈ K[x] has newly reducible nth iterate over K if fn−1 is irreducible over K, but fn is not (here fi denotes the ith iterate of f). We pose the problem of characterizing, for given d,n > 1, fields K such that there exists f ∈ K[x] of degree d with newly reducible nth iterate, and the similar problem for fields admitting infinitely many such f. We give results in the cases (d,n) ∈{(2, 3), (3, 2), (4, 2)} as well as for (d, 2) when d ≡ 2 mod 4. In particular, we show that for all these (d,n) pairs, there are infinitely many monic f ∈ ℤ[x] of degree d with newly reducible nth iterate over ℚ. Curiously, the minimal polynomial x2 − x − 1 of the golden ratio is one example of f ∈ ℤ[x] with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.

中文翻译：

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新可约多项式迭代

给定一个字段ķ和n > 1, 我们说一个多项式F ∈ ķ[X]有新约n迭代ķ如果Fn-1是不可约的ķ， 但Fn不在这里F一世表示一世的迭代F）。我们提出表征问题，因为给定d,n > 1, 字段ķ使得存在F ∈ ķ[X]学位d与新约n迭代，并且对于允许无限多这样的字段的类似问题F. 我们在案例中给出结果(d,n) ∈{(2, 3), (3, 2), (4, 2)}以及对于(d, 2)什么时候d ≡ 2模组 4. 特别是，我们表明对于所有这些(d,n)对，有无限多的monicF ∈ ℤ[X]学位d与新约n迭代ℚ. 奇怪的是，最小多项式X2 - X - 1黄金比例就是一个例子F ∈ ℤ[X]具有新可约的第三次迭代；很少有其他示例具有较小的系数。我们的调查引发了一些猜想和悬而未决的问题。