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An \(\ell-p\) switch trick to obtain a new proof of a criterion for arithmetic equivalence. (English) Zbl 1471.11261

Let \(K\) be an algebraic number field and \(\ell\) a rational prime. The arithmetic type \(A_K(\ell)\) of \(\ell\) in \(K\) is the sequence of degrees of prime ideal factors of \(\ell\) in \(K\), and the splitting number \(g_K(\ell)\) is the number of these factors. It has been proved by R. Perlis [J. Number Theory 9, 342–360 (1977; Zbl 0389.12006)] that two fields \(K,L\) are arithmetically equivalent, i.e., their Dedekind zeta-functions coincide, if and only if \[A_K(\ell)=A_L(\ell)\text{ for almost all } \ell.\tag{1}\] Later D. Stuart and R. Perlis [J. Number Theory 53, No. 2, 300–308 (1995; Zbl 0863.11082)] have shown that \(K\) and \(L\) are arithmetically equivalent if and only if \[g_K(\ell)=g_L(\ell)\text{ for almost all } \ell.\tag{2}\]
The authors present a short proof of the equivalence of the conditions (1) and (2), based on an elementary consequence of an old result of [H. J. S. Smith, Proc. Lond. Math. Soc. 7, 208–212 (1875; JFM 08.0074.03)].

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11R42 Zeta functions and \(L\)-functions of number fields
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References:

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