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On reciprocity equivalence of quadratic number fields. (English) Zbl 0733.11012
Two number fields K and L are said to be reciprocity equivalent if there exist a group isomorphism \(\phi: K^{\times}/K^{\times 2}\to L^{\times}/L^{\times 2}\) and a bijection \(\Phi: \Omega\) (K)\(\to \Omega (L)\) between the sets of all primes of K and L that preserve Hilbert symbols, i.e. \((a,b)_ P=(\phi a,\phi b)_{\Phi P}\) for all \(a,b\in K^{\times}/K^{\times 2}\) and \(P\in \Omega (K)\). If moreover \(ord_ P a\equiv ord_{\Phi P} \phi a (mod 2)\) for every finite \(P\in \Omega (K)\) and every \(a\in K^{\times}/K^{\times 2}\) then K and L are said to be tamely reciprocity equivalent. In the paper the author classifies quadratic number fields with respect to reciprocity equivalence and to tame reciprocity equivalence. The sufficient and necessary conditions for two quadratic number fields to be (tamely) reciprocity equivalent are formulated in terms of classical invariants, examination of which implies that there are exactly 7 classes of reciprocity equivalent quadratic number fields.
Reciprocity equivalence coincides with Witt equivalence [K. Szymiczek, Matching Witts locally and globally, Math. Slovaca 41, No.3, 315-331 (1991)], so it is worth pointing out that the results of this paper give also the classification of quadratic number fields with respect to their Witt rings.

11E12 Quadratic forms over global rings and fields
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11R11 Quadratic extensions
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