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, $\xi$ 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.