A polynomial $f(x)$ over a field K is called stable if all of its iterates are irreducible over K. In this paper, we study the stability of trinomials over finite fields. We show that if $f(x)$ is a trinomial of even degree over the binary field ${\mathbb{F}}_2$, then $f(x)$ is not stable. We prove similar results for some families of monic trinomials over finite fields of odd characteristic. We also study the stability of polynomials of higher weights and prove some results and pose a new conjecture.

中文翻译：

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关于有限域上的三项式稳定性的一个注记

多项式 $F（X）$在现场ķ被称为稳定的，如果所有的迭代都超过束缚ķ。在本文中，我们研究三项式在有限域上的稳定性。我们证明如果$F（X）$ 是二进制域上偶数次的三项式 ${\mathbb{F}}_2$， 然后 $F（X）$不稳定。我们证明了在奇特性有限域上的某些一元三次多项式族的相似结果。我们还研究了更高权重的多项式的稳定性，并证明了一些结果并提出了新的猜想。