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

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