A Generalization of Arithmetic Derivative to $p$-adic Fields and Number Fields

The arithmetic derivative is a function from the natural numbers to itself that sends all prime numbers to $1$ and satisfies the Leibniz rule. The arithmetic partial derivative with respect to a prime $p$ is the $p$-th component of the arithmetic derivative. In this paper, we generalize the arithmetic partial derivative to $p$-adic fields (the local case) and the arithmetic derivative to number fields (the global case). We study the dynamical system of the $p$-adic valuation of the iterations of the arithmetic partial derivatives. We also prove that for every integer $n\geq 0$, there are infinitely many elements with exactly $n$ anti-partial derivatives. In the end, we study the $p$-adic continuity of arithmetic derivatives.