On Geary's theorem for the field of p-adic numbers

Let a"e (p) , where p > 2, be a field of p-adic numbers. We consider two independent identically distributed random variables with values in a"e (p) and distribution mu with a continuous density. We prove that the sum and the squared difference of these random variables are independent if and only if mu is an idempotent distribution, i.e., a shift of the Haar distribution of a compact subgroup of the additive group of the field a"e (p) .