Abstract In this article, we study several probabilistic properties of polynomials defined over the ring of p-adic integers under the Haar measure. First, we calculate the probability that a monic polynomial is separable, generalizing a result of Polak. Second, we introduce the notion of two polynomials being strongly coprime and calculate the probability of two monic polynomials being strongly coprime. Finally, we explain how our method can be used to extrapolate other probabilistic properties of polynomials over the ring of p-adic integers from polynomials defined over the integers modulo powers of p.

中文翻译：

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p进整数环上多项式的某些概率性质

摘要 在本文中，我们研究了在 Haar 测度下在 p-adic 整数环上定义的多项式的几个概率特性。首先，我们计算单项多项式可分的概率，概括了 Polak 的结果。其次，我们引入两个多项式为强互质的概念，并计算两个单调多项式为强互质的概率。最后，我们解释了我们的方法如何用于从定义在 p 的整数模幂上的多项式推断 p-adic 整数环上多项式的其他概率特性。