  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.
