An $\mathrm{\ell }-p$ switch trick to obtain a new elementary proof of a criterion for arithmetic equivalence

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 $\mathrm{\ell }$ the arithmetic type of $\mathrm{\ell }$ 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 $\mathrm{\ell }$ to have the same number of prime factors in each field. Here, using an $\mathrm{\ell }-p$ switch trick, we provide an elementary proof of that fact based on a classical result of Smith from the 1870’s.