  Two number fields are called arithmetically equivalent if they have the same
Dedekind zeta function. In the 1970's Perlis showed that this is equivalent to
the condition that for almost every rational prime $\ell$ the arithmetic type
of $\ell$ is the same in each field. In the 1990's Perlis and Stuart gave an
unexpected characterization for arithmetic equivalence; they showed that to be
arithmetically equivalent it is enough for almost every prime $\ell$ to have
the same number of prime factors in each field. Here, using an $\ell-p$ switch
trick, we provide an elementary proof of that fact based on a classical result
of Smith from the 1870's.
