On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of p-Adic Numbers

We prove the following theorem. Let X be a discrete field, and (xi ) and (eta ) be independent identically distributed random variables with values in X and distribution (mu ). The random variables (S=xi +eta ) and (D=(xi -eta )^2) are independent if and only if (mu ) is an idempotent distribution. A similar result is also proved in the case when (xi ) and (eta ) are independent identically distributed random variables with values in the field of p-adic numbers ({mathbf {Q}}_p), where (p>2), assuming that the distribution (mu ) has a continuous density.