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 ≥ 0, there are infinitely many elements with exactly n anti-partial derivatives. In the end, we study the p-adic continuity of arithmetic derivatives.